Theorem fpwwe2lem6 10233

Description: Lemma for fpwwe2 10240. (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)
fpwwe2lem8.x	⊢ (𝜑 → 𝑋𝑊𝑅)
fpwwe2lem8.y	⊢ (𝜑 → 𝑌𝑊𝑆)
fpwwe2lem8.m	⊢ 𝑀 = OrdIso(𝑅, 𝑋)
fpwwe2lem8.n	⊢ 𝑁 = OrdIso(𝑆, 𝑌)
fpwwe2lem5.1	⊢ (𝜑 → 𝐵 ∈ dom 𝑀)
fpwwe2lem5.2	⊢ (𝜑 → 𝐵 ∈ dom 𝑁)
fpwwe2lem5.3	⊢ (𝜑 → (𝑀 ↾ 𝐵) = (𝑁 ↾ 𝐵))

Assertion

Ref	Expression
fpwwe2lem6	⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝐶𝑆(𝑁‘𝐵) ∧ (𝐷𝑅(𝑀‘𝐵) → (𝐶𝑅𝐷 ↔ 𝐶𝑆𝐷))))

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

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

Proof of Theorem fpwwe2lem6

Step	Hyp	Ref	Expression
1		fpwwe2lem8.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌𝑊𝑆)
2		fpwwe2.1	. . . . . . . . . 10 ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
3	2	relopabiv 5679	. . . . . . . . 9 ⊢ Rel 𝑊
4	3	brrelex1i 5594	. . . . . . . 8 ⊢ (𝑌𝑊𝑆 → 𝑌 ∈ V)
5	1, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ V)
6		fpwwe2.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7	2, 6	fpwwe2lem2 10229	. . . . . . . . 9 ⊢ (𝜑 → (𝑌𝑊𝑆 ↔ ((𝑌 ⊆ 𝐴 ∧ 𝑆 ⊆ (𝑌 × 𝑌)) ∧ (𝑆 We 𝑌 ∧ ∀𝑦 ∈ 𝑌 [(◡𝑆 “ {𝑦}) / 𝑢](𝑢𝐹(𝑆 ∩ (𝑢 × 𝑢))) = 𝑦))))
8	1, 7	mpbid 235	. . . . . . . 8 ⊢ (𝜑 → ((𝑌 ⊆ 𝐴 ∧ 𝑆 ⊆ (𝑌 × 𝑌)) ∧ (𝑆 We 𝑌 ∧ ∀𝑦 ∈ 𝑌 [(◡𝑆 “ {𝑦}) / 𝑢](𝑢𝐹(𝑆 ∩ (𝑢 × 𝑢))) = 𝑦)))
9	8	simprld 772	. . . . . . 7 ⊢ (𝜑 → 𝑆 We 𝑌)
10		fpwwe2lem8.n	. . . . . . . 8 ⊢ 𝑁 = OrdIso(𝑆, 𝑌)
11	10	oiiso 9142	. . . . . . 7 ⊢ ((𝑌 ∈ V ∧ 𝑆 We 𝑌) → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
12	5, 9, 11	syl2anc 587	. . . . . 6 ⊢ (𝜑 → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
13	12	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
14		isof1o 7121	. . . . 5 ⊢ (𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌) → 𝑁:dom 𝑁–1-1-onto→𝑌)
15	13, 14	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝑁:dom 𝑁–1-1-onto→𝑌)
16		fpwwe2.3	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
17		fpwwe2lem8.x	. . . . . 6 ⊢ (𝜑 → 𝑋𝑊𝑅)
18		fpwwe2lem8.m	. . . . . 6 ⊢ 𝑀 = OrdIso(𝑅, 𝑋)
19		fpwwe2lem5.1	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ dom 𝑀)
20		fpwwe2lem5.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ dom 𝑁)
21		fpwwe2lem5.3	. . . . . 6 ⊢ (𝜑 → (𝑀 ↾ 𝐵) = (𝑁 ↾ 𝐵))
22	2, 6, 16, 17, 1, 18, 10, 19, 20, 21	fpwwe2lem5 10232	. . . . 5 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝐶 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌 ∧ (◡𝑀‘𝐶) = (◡𝑁‘𝐶)))
23	22	simp2d 1145	. . . 4 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐶 ∈ 𝑌)
24		f1ocnvfv2 7077	. . . 4 ⊢ ((𝑁:dom 𝑁–1-1-onto→𝑌 ∧ 𝐶 ∈ 𝑌) → (𝑁‘(◡𝑁‘𝐶)) = 𝐶)
25	15, 23, 24	syl2anc 587	. . 3 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝑁‘(◡𝑁‘𝐶)) = 𝐶)
26	22	simp3d 1146	. . . . 5 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (◡𝑀‘𝐶) = (◡𝑁‘𝐶))
27	3	brrelex1i 5594	. . . . . . . . . . . 12 ⊢ (𝑋𝑊𝑅 → 𝑋 ∈ V)
28	17, 27	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ V)
29	2, 6	fpwwe2lem2 10229	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
30	17, 29	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
31	30	simprld 772	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 We 𝑋)
32	18	oiiso 9142	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ V ∧ 𝑅 We 𝑋) → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
33	28, 31, 32	syl2anc 587	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
34	33	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
35		isof1o 7121	. . . . . . . . 9 ⊢ (𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋) → 𝑀:dom 𝑀–1-1-onto→𝑋)
36	34, 35	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝑀:dom 𝑀–1-1-onto→𝑋)
37	22	simp1d 1144	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐶 ∈ 𝑋)
38		f1ocnvfv2 7077	. . . . . . . 8 ⊢ ((𝑀:dom 𝑀–1-1-onto→𝑋 ∧ 𝐶 ∈ 𝑋) → (𝑀‘(◡𝑀‘𝐶)) = 𝐶)
39	36, 37, 38	syl2anc 587	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝑀‘(◡𝑀‘𝐶)) = 𝐶)
40		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐶𝑅(𝑀‘𝐵))
41	39, 40	eqbrtrd 5065	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝑀‘(◡𝑀‘𝐶))𝑅(𝑀‘𝐵))
42		f1ocnv 6662	. . . . . . . . 9 ⊢ (𝑀:dom 𝑀–1-1-onto→𝑋 → ◡𝑀:𝑋–1-1-onto→dom 𝑀)
43		f1of 6650	. . . . . . . . 9 ⊢ (◡𝑀:𝑋–1-1-onto→dom 𝑀 → ◡𝑀:𝑋⟶dom 𝑀)
44	36, 42, 43	3syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ◡𝑀:𝑋⟶dom 𝑀)
45	44, 37	ffvelrnd 6894	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (◡𝑀‘𝐶) ∈ dom 𝑀)
46	19	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐵 ∈ dom 𝑀)
47		isorel 7124	. . . . . . 7 ⊢ ((𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋) ∧ ((◡𝑀‘𝐶) ∈ dom 𝑀 ∧ 𝐵 ∈ dom 𝑀)) → ((◡𝑀‘𝐶) E 𝐵 ↔ (𝑀‘(◡𝑀‘𝐶))𝑅(𝑀‘𝐵)))
48	34, 45, 46, 47	syl12anc 837	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ((◡𝑀‘𝐶) E 𝐵 ↔ (𝑀‘(◡𝑀‘𝐶))𝑅(𝑀‘𝐵)))
49	41, 48	mpbird 260	. . . . 5 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (◡𝑀‘𝐶) E 𝐵)
50	26, 49	eqbrtrrd 5067	. . . 4 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (◡𝑁‘𝐶) E 𝐵)
51		f1ocnv 6662	. . . . . . 7 ⊢ (𝑁:dom 𝑁–1-1-onto→𝑌 → ◡𝑁:𝑌–1-1-onto→dom 𝑁)
52		f1of 6650	. . . . . . 7 ⊢ (◡𝑁:𝑌–1-1-onto→dom 𝑁 → ◡𝑁:𝑌⟶dom 𝑁)
53	15, 51, 52	3syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ◡𝑁:𝑌⟶dom 𝑁)
54	53, 23	ffvelrnd 6894	. . . . 5 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (◡𝑁‘𝐶) ∈ dom 𝑁)
55	20	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐵 ∈ dom 𝑁)
56		isorel 7124	. . . . 5 ⊢ ((𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌) ∧ ((◡𝑁‘𝐶) ∈ dom 𝑁 ∧ 𝐵 ∈ dom 𝑁)) → ((◡𝑁‘𝐶) E 𝐵 ↔ (𝑁‘(◡𝑁‘𝐶))𝑆(𝑁‘𝐵)))
57	13, 54, 55, 56	syl12anc 837	. . . 4 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → ((◡𝑁‘𝐶) E 𝐵 ↔ (𝑁‘(◡𝑁‘𝐶))𝑆(𝑁‘𝐵)))
58	50, 57	mpbid 235	. . 3 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝑁‘(◡𝑁‘𝐶))𝑆(𝑁‘𝐵))
59	25, 58	eqbrtrrd 5067	. 2 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → 𝐶𝑆(𝑁‘𝐵))
60	26	adantrr 717	. . . . 5 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → (◡𝑀‘𝐶) = (◡𝑁‘𝐶))
61	2, 6, 16, 17, 1, 18, 10, 19, 20, 21	fpwwe2lem5 10232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐷𝑅(𝑀‘𝐵)) → (𝐷 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌 ∧ (◡𝑀‘𝐷) = (◡𝑁‘𝐷)))
62	61	simp3d 1146	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷𝑅(𝑀‘𝐵)) → (◡𝑀‘𝐷) = (◡𝑁‘𝐷))
63	62	adantrl 716	. . . . 5 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → (◡𝑀‘𝐷) = (◡𝑁‘𝐷))
64	60, 63	breq12d 5056	. . . 4 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → ((◡𝑀‘𝐶) E (◡𝑀‘𝐷) ↔ (◡𝑁‘𝐶) E (◡𝑁‘𝐷)))
65	33	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → 𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋))
66		isocnv 7128	. . . . . 6 ⊢ (𝑀 Isom E , 𝑅 (dom 𝑀, 𝑋) → ◡𝑀 Isom 𝑅, E (𝑋, dom 𝑀))
67	65, 66	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → ◡𝑀 Isom 𝑅, E (𝑋, dom 𝑀))
68	37	adantrr 717	. . . . 5 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → 𝐶 ∈ 𝑋)
69	30	simplrd 770	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ⊆ (𝑋 × 𝑋))
70	69	ssbrd 5086	. . . . . . . 8 ⊢ (𝜑 → (𝐷𝑅(𝑀‘𝐵) → 𝐷(𝑋 × 𝑋)(𝑀‘𝐵)))
71	70	imp 410	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐷𝑅(𝑀‘𝐵)) → 𝐷(𝑋 × 𝑋)(𝑀‘𝐵))
72		brxp 5587	. . . . . . . 8 ⊢ (𝐷(𝑋 × 𝑋)(𝑀‘𝐵) ↔ (𝐷 ∈ 𝑋 ∧ (𝑀‘𝐵) ∈ 𝑋))
73	72	simplbi 501	. . . . . . 7 ⊢ (𝐷(𝑋 × 𝑋)(𝑀‘𝐵) → 𝐷 ∈ 𝑋)
74	71, 73	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷𝑅(𝑀‘𝐵)) → 𝐷 ∈ 𝑋)
75	74	adantrl 716	. . . . 5 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → 𝐷 ∈ 𝑋)
76		isorel 7124	. . . . 5 ⊢ ((◡𝑀 Isom 𝑅, E (𝑋, dom 𝑀) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (𝐶𝑅𝐷 ↔ (◡𝑀‘𝐶) E (◡𝑀‘𝐷)))
77	67, 68, 75, 76	syl12anc 837	. . . 4 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → (𝐶𝑅𝐷 ↔ (◡𝑀‘𝐶) E (◡𝑀‘𝐷)))
78	12	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → 𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌))
79		isocnv 7128	. . . . . 6 ⊢ (𝑁 Isom E , 𝑆 (dom 𝑁, 𝑌) → ◡𝑁 Isom 𝑆, E (𝑌, dom 𝑁))
80	78, 79	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → ◡𝑁 Isom 𝑆, E (𝑌, dom 𝑁))
81	23	adantrr 717	. . . . 5 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → 𝐶 ∈ 𝑌)
82	61	simp2d 1145	. . . . . 6 ⊢ ((𝜑 ∧ 𝐷𝑅(𝑀‘𝐵)) → 𝐷 ∈ 𝑌)
83	82	adantrl 716	. . . . 5 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → 𝐷 ∈ 𝑌)
84		isorel 7124	. . . . 5 ⊢ ((◡𝑁 Isom 𝑆, E (𝑌, dom 𝑁) ∧ (𝐶 ∈ 𝑌 ∧ 𝐷 ∈ 𝑌)) → (𝐶𝑆𝐷 ↔ (◡𝑁‘𝐶) E (◡𝑁‘𝐷)))
85	80, 81, 83, 84	syl12anc 837	. . . 4 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → (𝐶𝑆𝐷 ↔ (◡𝑁‘𝐶) E (◡𝑁‘𝐷)))
86	64, 77, 85	3bitr4d 314	. . 3 ⊢ ((𝜑 ∧ (𝐶𝑅(𝑀‘𝐵) ∧ 𝐷𝑅(𝑀‘𝐵))) → (𝐶𝑅𝐷 ↔ 𝐶𝑆𝐷))
87	86	expr 460	. 2 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝐷𝑅(𝑀‘𝐵) → (𝐶𝑅𝐷 ↔ 𝐶𝑆𝐷)))
88	59, 87	jca 515	1 ⊢ ((𝜑 ∧ 𝐶𝑅(𝑀‘𝐵)) → (𝐶𝑆(𝑁‘𝐵) ∧ (𝐷𝑅(𝑀‘𝐵) → (𝐶𝑅𝐷 ↔ 𝐶𝑆𝐷))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∀wral 3054  Vcvv 3401  [wsbc 3687   ∩ cin 3856   ⊆ wss 3857  {csn 4531   class class class wbr 5043  {copab 5105   E cep 5448   We wwe 5497   × cxp 5538  ^◡ccnv 5539  dom cdm 5540   ↾ cres 5542   “ cima 5543  ⟶wf 6365  –1-1-onto→wf1o 6368  ‘cfv 6369   Isom wiso 6370  (class class class)co 7202  OrdIsocoi 9114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706  ax-rep 5168  ax-sep 5181  ax-nul 5188  ax-pow 5247  ax-pr 5311  ax-un 7512

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2537  df-eu 2566  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-pw 4505  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-iun 4896  df-br 5044  df-opab 5106  df-mpt 5125  df-tr 5151  df-id 5444  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-se 5499  df-we 5500  df-xp 5546  df-rel 5547  df-cnv 5548  df-co 5549  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149  df-ord 6205  df-on 6206  df-lim 6207  df-suc 6208  df-iota 6327  df-fun 6371  df-fn 6372  df-f 6373  df-f1 6374  df-fo 6375  df-f1o 6376  df-fv 6377  df-isom 6378  df-riota 7159  df-ov 7205  df-wrecs 8036  df-recs 8097  df-oi 9115

This theorem is referenced by:  fpwwe2lem7  10234