Theorem fpwwe2 10623

Description: Given any function 𝐹 from well-orderings of subsets of 𝐴 to 𝐴, there is a unique well-ordered subset ⟨𝑋, (𝑊‘𝑋)⟩ which "agrees" with 𝐹 in the sense that each initial segment maps to its upper bound, and such that the entire set maps to an element of the set (so that it cannot be extended without losing the well-ordering). This theorem can be used to prove dfac8a 10010. Theorem 1.1 of [KanamoriPincus] p. 415. (Contributed by Mario Carneiro, 18-May-2015.) (Revised by AV, 20-Jul-2024.)

Hypotheses

Ref	Expression
fpwwe2.1	⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
fpwwe2.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fpwwe2.3	⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
fpwwe2.4	⊢ 𝑋 = ∪ dom 𝑊

Assertion

Ref	Expression
fpwwe2	⊢ (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) ↔ (𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋))))

Distinct variable groups:   𝑦,𝑢,𝑟,𝑥,𝐹   𝑋,𝑟,𝑢,𝑥,𝑦   𝜑,𝑟,𝑢,𝑥,𝑦   𝐴,𝑟,𝑥   𝑅,𝑟,𝑢,𝑥,𝑦   𝑌,𝑟,𝑢,𝑥,𝑦   𝑊,𝑟,𝑢,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑦,𝑢)   𝑉(𝑥,𝑦,𝑢,𝑟)

Proof of Theorem fpwwe2

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fpwwe2.1	. . . . . . . . . . 11 ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
2		fpwwe2.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		fpwwe2.3	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
4		fpwwe2.4	. . . . . . . . . . 11 ⊢ 𝑋 = ∪ dom 𝑊
5	1, 2, 3, 4	fpwwe2lem10 10620	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋))
6	5	ffund 6710	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝑊)
7		funbrfv2b 6938	. . . . . . . . 9 ⊢ (Fun 𝑊 → (𝑌𝑊𝑅 ↔ (𝑌 ∈ dom 𝑊 ∧ (𝑊‘𝑌) = 𝑅)))
8	6, 7	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝑌𝑊𝑅 ↔ (𝑌 ∈ dom 𝑊 ∧ (𝑊‘𝑌) = 𝑅)))
9	8	simprbda 503	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌𝑊𝑅) → 𝑌 ∈ dom 𝑊)
10	9	adantrr 729	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌 ∈ dom 𝑊)
11		elssuni 4904	. . . . . . 7 ⊢ (𝑌 ∈ dom 𝑊 → 𝑌 ⊆ ∪ dom 𝑊)
12	11, 4	sseqtrrdi 3978	. . . . . 6 ⊢ (𝑌 ∈ dom 𝑊 → 𝑌 ⊆ 𝑋)
13	10, 12	syl 18	. . . . 5 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌 ⊆ 𝑋)
14		simpl 487	. . . . . . 7 ⊢ ((𝑋 ⊆ 𝑌 ∧ (𝑊‘𝑋) = (𝑅 ∩ (𝑌 × 𝑋))) → 𝑋 ⊆ 𝑌)
15	14	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑋 ⊆ 𝑌 ∧ (𝑊‘𝑋) = (𝑅 ∩ (𝑌 × 𝑋))) → 𝑋 ⊆ 𝑌))
16		simplrr 789	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑌𝐹𝑅) ∈ 𝑌)
17	2	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝐴 ∈ 𝑉)
18	17	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → 𝐴 ∈ 𝑉)
19	1, 2, 3, 4	fpwwe2lem11 10621	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑋 ∈ dom 𝑊)
20		funfvbrb 7046	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Fun 𝑊 → (𝑋 ∈ dom 𝑊 ↔ 𝑋𝑊(𝑊‘𝑋)))
21	6, 20	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑋 ∈ dom 𝑊 ↔ 𝑋𝑊(𝑊‘𝑋)))
22	19, 21	mpbid 235	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑋𝑊(𝑊‘𝑋))
23	1, 2	fpwwe2lem2 10612	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑋𝑊(𝑊‘𝑋) ↔ ((𝑋 ⊆ 𝐴 ∧ (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)) ∧ ((𝑊‘𝑋) We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡(𝑊‘𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊‘𝑋) ∩ (𝑢 × 𝑢))) = 𝑦))))
24	22, 23	mpbid 235	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑋 ⊆ 𝐴 ∧ (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)) ∧ ((𝑊‘𝑋) We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡(𝑊‘𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊‘𝑋) ∩ (𝑢 × 𝑢))) = 𝑦)))
25	24	ad2antrr 738	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋 ⊆ 𝐴 ∧ (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)) ∧ ((𝑊‘𝑋) We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡(𝑊‘𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊‘𝑋) ∩ (𝑢 × 𝑢))) = 𝑦)))
26	25	simpld 499	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋 ⊆ 𝐴 ∧ (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)))
27	26	simpld 499	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋 ⊆ 𝐴)
28	18, 27	ssexd 5295	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋 ∈ V)
29	28	difexd 5302	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋 ∖ 𝑌) ∈ V)
30	25	simprd 500	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑊‘𝑋) We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡(𝑊‘𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊‘𝑋) ∩ (𝑢 × 𝑢))) = 𝑦))
31	30	simpld 499	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑊‘𝑋) We 𝑋)
32		wefr 5651	. . . . . . . . . . . . 13 ⊢ ((𝑊‘𝑋) We 𝑋 → (𝑊‘𝑋) Fr 𝑋)
33	31, 32	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑊‘𝑋) Fr 𝑋)
34		difssd 4091	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋 ∖ 𝑌) ⊆ 𝑋)
35		fri 5619	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∖ 𝑌) ∈ V ∧ (𝑊‘𝑋) Fr 𝑋) ∧ ((𝑋 ∖ 𝑌) ⊆ 𝑋 ∧ (𝑋 ∖ 𝑌) ≠ ∅)) → ∃𝑧 ∈ (𝑋 ∖ 𝑌)∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧)
36	35	expr 461	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∖ 𝑌) ∈ V ∧ (𝑊‘𝑋) Fr 𝑋) ∧ (𝑋 ∖ 𝑌) ⊆ 𝑋) → ((𝑋 ∖ 𝑌) ≠ ∅ → ∃𝑧 ∈ (𝑋 ∖ 𝑌)∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧))
37	29, 33, 34, 36	syl21anc 850	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋 ∖ 𝑌) ≠ ∅ → ∃𝑧 ∈ (𝑋 ∖ 𝑌)∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧))
38		ssdif0 4321	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) ⊆ 𝑌 ↔ ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) ∖ 𝑌) = ∅)
39		indif1 4235	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∖ 𝑌) ∩ (◡(𝑊‘𝑋) “ {𝑧})) = ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) ∖ 𝑌)
40	39	eqeq1i 2768	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∖ 𝑌) ∩ (◡(𝑊‘𝑋) “ {𝑧})) = ∅ ↔ ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) ∖ 𝑌) = ∅)
41		disj 4410	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∖ 𝑌) ∩ (◡(𝑊‘𝑋) “ {𝑧})) = ∅ ↔ ∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤 ∈ (◡(𝑊‘𝑋) “ {𝑧}))
42		vex 3459	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑤 ∈ V
43	42	eliniseg 6096	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ V → (𝑤 ∈ (◡(𝑊‘𝑋) “ {𝑧}) ↔ 𝑤(𝑊‘𝑋)𝑧))
44	43	elv 3460	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 ∈ (◡(𝑊‘𝑋) “ {𝑧}) ↔ 𝑤(𝑊‘𝑋)𝑧)
45	44	notbii 323	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑤 ∈ (◡(𝑊‘𝑋) “ {𝑧}) ↔ ¬ 𝑤(𝑊‘𝑋)𝑧)
46	45	ralbii 3111	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤 ∈ (◡(𝑊‘𝑋) “ {𝑧}) ↔ ∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧)
47	41, 46	bitri 278	. . . . . . . . . . . . . . 15 ⊢ (((𝑋 ∖ 𝑌) ∩ (◡(𝑊‘𝑋) “ {𝑧})) = ∅ ↔ ∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧)
48	38, 40, 47	3bitr2i 302	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) ⊆ 𝑌 ↔ ∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧)
49		cnvimass 6084	. . . . . . . . . . . . . . . . 17 ⊢ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ dom (𝑊‘𝑋)
50	26	simprd 500	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑊‘𝑋) ⊆ (𝑋 × 𝑋))
51		dmss 5892	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑊‘𝑋) ⊆ (𝑋 × 𝑋) → dom (𝑊‘𝑋) ⊆ dom (𝑋 × 𝑋))
52	50, 51	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → dom (𝑊‘𝑋) ⊆ dom (𝑋 × 𝑋))
53		dmxpid 5920	. . . . . . . . . . . . . . . . . 18 ⊢ dom (𝑋 × 𝑋) = 𝑋
54	52, 53	sseqtrdi 3977	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → dom (𝑊‘𝑋) ⊆ 𝑋)
55	49, 54	sstrid 3948	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑋)
56		sseqin2 4176	. . . . . . . . . . . . . . . 16 ⊢ ((◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑋 ↔ (𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) = (◡(𝑊‘𝑋) “ {𝑧}))
57	55, 56	sylib 221	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) = (◡(𝑊‘𝑋) “ {𝑧}))
58	57	sseq1d 3968	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) ⊆ 𝑌 ↔ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌))
59	48, 58	bitr3id 288	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧 ↔ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌))
60	59	rexbidv 3189	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (∃𝑧 ∈ (𝑋 ∖ 𝑌)∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧 ↔ ∃𝑧 ∈ (𝑋 ∖ 𝑌)(◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌))
61		eldifn 4086	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 ∈ (𝑋 ∖ 𝑌) → ¬ 𝑧 ∈ 𝑌)
62	61	ad2antrl 740	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → ¬ 𝑧 ∈ 𝑌)
63		eleq1w 2846	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝑌 ↔ 𝑧 ∈ 𝑌))
64	63	notbid 321	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑤 = 𝑧 → (¬ 𝑤 ∈ 𝑌 ↔ ¬ 𝑧 ∈ 𝑌))
65	62, 64	syl5ibrcom 250	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑤 = 𝑧 → ¬ 𝑤 ∈ 𝑌))
66	65	con2d 135	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑤 ∈ 𝑌 → ¬ 𝑤 = 𝑧))
67	66	imp 411	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → ¬ 𝑤 = 𝑧)
68	62	adantr 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → ¬ 𝑧 ∈ 𝑌)
69		simprr 784	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))
70	69	ad2antrr 738	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))
71	70	breqd 5120	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑧𝑅𝑤 ↔ 𝑧((𝑊‘𝑋) ∩ (𝑋 × 𝑌))𝑤))
72		eldifi 4085	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑧 ∈ (𝑋 ∖ 𝑌) → 𝑧 ∈ 𝑋)
73	72	ad2antrl 740	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑧 ∈ 𝑋)
74	73	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → 𝑧 ∈ 𝑋)
75		simpr 489	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ 𝑌)
76		brxp 5710	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧(𝑋 × 𝑌)𝑤 ↔ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌))
77	74, 75, 76	sylanbrc 594	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → 𝑧(𝑋 × 𝑌)𝑤)
78		brin 5163	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑧((𝑊‘𝑋) ∩ (𝑋 × 𝑌))𝑤 ↔ (𝑧(𝑊‘𝑋)𝑤 ∧ 𝑧(𝑋 × 𝑌)𝑤))
79	78	rbaib 547	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧(𝑋 × 𝑌)𝑤 → (𝑧((𝑊‘𝑋) ∩ (𝑋 × 𝑌))𝑤 ↔ 𝑧(𝑊‘𝑋)𝑤))
80	77, 79	syl 18	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑧((𝑊‘𝑋) ∩ (𝑋 × 𝑌))𝑤 ↔ 𝑧(𝑊‘𝑋)𝑤))
81	71, 80	bitrd 282	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑧𝑅𝑤 ↔ 𝑧(𝑊‘𝑋)𝑤))
82	1, 2	fpwwe2lem2 10612	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → (𝑌𝑊𝑅 ↔ ((𝑌 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑌 × 𝑌)) ∧ (𝑅 We 𝑌 ∧ ∀𝑦 ∈ 𝑌 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
83	82	biimpa 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑌𝑊𝑅) → ((𝑌 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑌 × 𝑌)) ∧ (𝑅 We 𝑌 ∧ ∀𝑦 ∈ 𝑌 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
84	83	adantrr 729	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑌 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑌 × 𝑌)) ∧ (𝑅 We 𝑌 ∧ ∀𝑦 ∈ 𝑌 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
85	84	simpld 499	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → (𝑌 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑌 × 𝑌)))
86	85	simprd 500	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑅 ⊆ (𝑌 × 𝑌))
87	86	ad3antrrr 742	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → 𝑅 ⊆ (𝑌 × 𝑌))
88	87	ssbrd 5154	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑧𝑅𝑤 → 𝑧(𝑌 × 𝑌)𝑤))
89		brxp 5710	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧(𝑌 × 𝑌)𝑤 ↔ (𝑧 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌))
90	89	simplbi 501	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧(𝑌 × 𝑌)𝑤 → 𝑧 ∈ 𝑌)
91	88, 90	syl6 36	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑧𝑅𝑤 → 𝑧 ∈ 𝑌))
92	81, 91	sylbird 263	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑧(𝑊‘𝑋)𝑤 → 𝑧 ∈ 𝑌))
93	68, 92	mtod 201	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → ¬ 𝑧(𝑊‘𝑋)𝑤)
94	31	ad2antrr 738	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑊‘𝑋) We 𝑋)
95		weso 5652	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑊‘𝑋) We 𝑋 → (𝑊‘𝑋) Or 𝑋)
96	94, 95	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑊‘𝑋) Or 𝑋)
97	13	ad2antrr 738	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑌 ⊆ 𝑋)
98	97	sselda 3937	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ 𝑋)
99		sotric 5599	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑊‘𝑋) Or 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑤(𝑊‘𝑋)𝑧 ↔ ¬ (𝑤 = 𝑧 ∨ 𝑧(𝑊‘𝑋)𝑤)))
100		ioran 999	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (¬ (𝑤 = 𝑧 ∨ 𝑧(𝑊‘𝑋)𝑤) ↔ (¬ 𝑤 = 𝑧 ∧ ¬ 𝑧(𝑊‘𝑋)𝑤))
101	99, 100	bitrdi 290	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑊‘𝑋) Or 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑤(𝑊‘𝑋)𝑧 ↔ (¬ 𝑤 = 𝑧 ∧ ¬ 𝑧(𝑊‘𝑋)𝑤)))
102	96, 98, 74, 101	syl12anc 849	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑤(𝑊‘𝑋)𝑧 ↔ (¬ 𝑤 = 𝑧 ∧ ¬ 𝑧(𝑊‘𝑋)𝑤)))
103	67, 93, 102	mpbir2and 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → 𝑤(𝑊‘𝑋)𝑧)
104	103, 44	sylibr 237	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ (◡(𝑊‘𝑋) “ {𝑧}))
105	104	ex 417	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑤 ∈ 𝑌 → 𝑤 ∈ (◡(𝑊‘𝑋) “ {𝑧})))
106	105	ssrdv 3943	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑌 ⊆ (◡(𝑊‘𝑋) “ {𝑧}))
107		simprr 784	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)
108	106, 107	eqssd 3954	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑌 = (◡(𝑊‘𝑋) “ {𝑧}))
109		in32 4182	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑊‘𝑋) ∩ (𝑋 × 𝑌)) ∩ (𝑌 × 𝑌)) = (((𝑊‘𝑋) ∩ (𝑌 × 𝑌)) ∩ (𝑋 × 𝑌))
110		simplrr 789	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))
111	110	ineq1d 4172	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑅 ∩ (𝑌 × 𝑌)) = (((𝑊‘𝑋) ∩ (𝑋 × 𝑌)) ∩ (𝑌 × 𝑌)))
112	86	ad2antrr 738	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 ⊆ (𝑌 × 𝑌))
113		dfss2 3923	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑅 ⊆ (𝑌 × 𝑌) ↔ (𝑅 ∩ (𝑌 × 𝑌)) = 𝑅)
114	112, 113	sylib 221	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑅 ∩ (𝑌 × 𝑌)) = 𝑅)
115	111, 114	eqtr3d 2800	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (((𝑊‘𝑋) ∩ (𝑋 × 𝑌)) ∩ (𝑌 × 𝑌)) = 𝑅)
116		inss2 4190	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑊‘𝑋) ∩ (𝑌 × 𝑌)) ⊆ (𝑌 × 𝑌)
117		xpss1 5680	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑌 ⊆ 𝑋 → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑌))
118	97, 117	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑌))
119	116, 118	sstrid 3948	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → ((𝑊‘𝑋) ∩ (𝑌 × 𝑌)) ⊆ (𝑋 × 𝑌))
120		dfss2 3923	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑊‘𝑋) ∩ (𝑌 × 𝑌)) ⊆ (𝑋 × 𝑌) ↔ (((𝑊‘𝑋) ∩ (𝑌 × 𝑌)) ∩ (𝑋 × 𝑌)) = ((𝑊‘𝑋) ∩ (𝑌 × 𝑌)))
121	119, 120	sylib 221	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (((𝑊‘𝑋) ∩ (𝑌 × 𝑌)) ∩ (𝑋 × 𝑌)) = ((𝑊‘𝑋) ∩ (𝑌 × 𝑌)))
122	109, 115, 121	3eqtr3a 2822	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 = ((𝑊‘𝑋) ∩ (𝑌 × 𝑌)))
123	108	sqxpeqd 5693	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌 × 𝑌) = ((◡(𝑊‘𝑋) “ {𝑧}) × (◡(𝑊‘𝑋) “ {𝑧})))
124	123	ineq2d 4173	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → ((𝑊‘𝑋) ∩ (𝑌 × 𝑌)) = ((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑧}) × (◡(𝑊‘𝑋) “ {𝑧}))))
125	122, 124	eqtrd 2798	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 = ((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑧}) × (◡(𝑊‘𝑋) “ {𝑧}))))
126	108, 125	oveq12d 7428	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌𝐹𝑅) = ((◡(𝑊‘𝑋) “ {𝑧})𝐹((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑧}) × (◡(𝑊‘𝑋) “ {𝑧})))))
127	18	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝐴 ∈ 𝑉)
128	22	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋𝑊(𝑊‘𝑋))
129	128	ad2antrr 738	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑋𝑊(𝑊‘𝑋))
130	1, 127, 129	fpwwe2lem3 10613	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑧 ∈ 𝑋) → ((◡(𝑊‘𝑋) “ {𝑧})𝐹((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑧}) × (◡(𝑊‘𝑋) “ {𝑧})))) = 𝑧)
131	73, 130	mpdan 699	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → ((◡(𝑊‘𝑋) “ {𝑧})𝐹((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑧}) × (◡(𝑊‘𝑋) “ {𝑧})))) = 𝑧)
132	126, 131	eqtrd 2798	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌𝐹𝑅) = 𝑧)
133	132, 62	eqneltrd 2883	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → ¬ (𝑌𝐹𝑅) ∈ 𝑌)
134	133	rexlimdvaa 3167	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (∃𝑧 ∈ (𝑋 ∖ 𝑌)(◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌 → ¬ (𝑌𝐹𝑅) ∈ 𝑌))
135	60, 134	sylbid 243	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (∃𝑧 ∈ (𝑋 ∖ 𝑌)∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧 → ¬ (𝑌𝐹𝑅) ∈ 𝑌))
136	37, 135	syld 48	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋 ∖ 𝑌) ≠ ∅ → ¬ (𝑌𝐹𝑅) ∈ 𝑌))
137	136	necon4ad 2977	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑌𝐹𝑅) ∈ 𝑌 → (𝑋 ∖ 𝑌) = ∅))
138	16, 137	mpd 16	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋 ∖ 𝑌) = ∅)
139		ssdif0 4321	. . . . . . . 8 ⊢ (𝑋 ⊆ 𝑌 ↔ (𝑋 ∖ 𝑌) = ∅)
140	138, 139	sylibr 237	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋 ⊆ 𝑌)
141	140	ex 417	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌))) → 𝑋 ⊆ 𝑌))
142	3	adantlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
143		simprl 782	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌𝑊𝑅)
144	1, 17, 142, 128, 143	fpwwe2lem9 10619	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑋 ⊆ 𝑌 ∧ (𝑊‘𝑋) = (𝑅 ∩ (𝑌 × 𝑋))) ∨ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))))
145	15, 141, 144	mpjaod 873	. . . . 5 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋 ⊆ 𝑌)
146	13, 145	eqssd 3954	. . . 4 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌 = 𝑋)
147	6	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → Fun 𝑊)
148	146, 143	eqbrtrrd 5135	. . . . . 6 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋𝑊𝑅)
149		funbrfv 6929	. . . . . 6 ⊢ (Fun 𝑊 → (𝑋𝑊𝑅 → (𝑊‘𝑋) = 𝑅))
150	147, 148, 149	sylc 66	. . . . 5 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → (𝑊‘𝑋) = 𝑅)
151	150	eqcomd 2769	. . . 4 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑅 = (𝑊‘𝑋))
152	146, 151	jca 520	. . 3 ⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → (𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)))
153	152	ex 417	. 2 ⊢ (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) → (𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋))))
154	1, 2, 3, 4	fpwwe2lem12 10622	. . . 4 ⊢ (𝜑 → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋)
155	22, 154	jca 520	. . 3 ⊢ (𝜑 → (𝑋𝑊(𝑊‘𝑋) ∧ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋))
156		breq12 5114	. . . 4 ⊢ ((𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)) → (𝑌𝑊𝑅 ↔ 𝑋𝑊(𝑊‘𝑋)))
157		oveq12 7419	. . . . 5 ⊢ ((𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)) → (𝑌𝐹𝑅) = (𝑋𝐹(𝑊‘𝑋)))
158		simpl 487	. . . . 5 ⊢ ((𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)) → 𝑌 = 𝑋)
159	157, 158	eleq12d 2857	. . . 4 ⊢ ((𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)) → ((𝑌𝐹𝑅) ∈ 𝑌 ↔ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋))
160	156, 159	anbi12d 643	. . 3 ⊢ ((𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)) → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) ↔ (𝑋𝑊(𝑊‘𝑋) ∧ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋)))
161	155, 160	syl5ibrcom 250	. 2 ⊢ (𝜑 → ((𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)) → (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)))
162	153, 161	impbid 215	1 ⊢ (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) ↔ (𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  Vcvv 3455  [wsbc 3744   ∖ cdif 3902   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  𝒫 cpw 4562  {csn 4589  ∪ cuni 4872   class class class wbr 5109  {copab 5173   Or wor 5568   Fr wfr 5611   We wwe 5613   × cxp 5659  ^◡ccnv 5660  dom cdm 5661   “ cima 5664  Fun wfun 6530  ‘cfv 6536  (class class class)co 7410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-oi 9468

This theorem is referenced by:  fpwwe  10626  canthwelem  10630  pwfseqlem4  10642