Theorem fpwwe2lem10 10254

Description: Lemma for fpwwe2 10257. (Contributed by Mario Carneiro, 15-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
fpwwe2lem10	⊢ (𝜑 → 𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋))

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

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

Proof of Theorem fpwwe2lem10

Dummy variables 𝑠 𝑡 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fpwwe2.1	. . . . . 6 ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
2	1	relopabiv 5690	. . . . 5 ⊢ Rel 𝑊
3	2	a1i 11	. . . 4 ⊢ (𝜑 → Rel 𝑊)
4		simprr 773	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) ∧ (𝑤 ⊆ 𝑤 ∧ 𝑠 = (𝑡 ∩ (𝑤 × 𝑤)))) → 𝑠 = (𝑡 ∩ (𝑤 × 𝑤)))
5		fpwwe2.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6	1, 5	fpwwe2lem2 10246	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑤𝑊𝑡 ↔ ((𝑤 ⊆ 𝐴 ∧ 𝑡 ⊆ (𝑤 × 𝑤)) ∧ (𝑡 We 𝑤 ∧ ∀𝑦 ∈ 𝑤 [(◡𝑡 “ {𝑦}) / 𝑢](𝑢𝐹(𝑡 ∩ (𝑢 × 𝑢))) = 𝑦))))
7	6	simprbda 502	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤𝑊𝑡) → (𝑤 ⊆ 𝐴 ∧ 𝑡 ⊆ (𝑤 × 𝑤)))
8	7	simprd 499	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤𝑊𝑡) → 𝑡 ⊆ (𝑤 × 𝑤))
9	8	adantrl 716	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) → 𝑡 ⊆ (𝑤 × 𝑤))
10	9	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) ∧ (𝑤 ⊆ 𝑤 ∧ 𝑠 = (𝑡 ∩ (𝑤 × 𝑤)))) → 𝑡 ⊆ (𝑤 × 𝑤))
11		df-ss 3883	. . . . . . . . . 10 ⊢ (𝑡 ⊆ (𝑤 × 𝑤) ↔ (𝑡 ∩ (𝑤 × 𝑤)) = 𝑡)
12	10, 11	sylib 221	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) ∧ (𝑤 ⊆ 𝑤 ∧ 𝑠 = (𝑡 ∩ (𝑤 × 𝑤)))) → (𝑡 ∩ (𝑤 × 𝑤)) = 𝑡)
13	4, 12	eqtrd 2777	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) ∧ (𝑤 ⊆ 𝑤 ∧ 𝑠 = (𝑡 ∩ (𝑤 × 𝑤)))) → 𝑠 = 𝑡)
14		simprr 773	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) ∧ (𝑤 ⊆ 𝑤 ∧ 𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))) → 𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))
15	1, 5	fpwwe2lem2 10246	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑤𝑊𝑠 ↔ ((𝑤 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑤 × 𝑤)) ∧ (𝑠 We 𝑤 ∧ ∀𝑦 ∈ 𝑤 [(◡𝑠 “ {𝑦}) / 𝑢](𝑢𝐹(𝑠 ∩ (𝑢 × 𝑢))) = 𝑦))))
16	15	simprbda 502	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤𝑊𝑠) → (𝑤 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑤 × 𝑤)))
17	16	simprd 499	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤𝑊𝑠) → 𝑠 ⊆ (𝑤 × 𝑤))
18	17	adantrr 717	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) → 𝑠 ⊆ (𝑤 × 𝑤))
19	18	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) ∧ (𝑤 ⊆ 𝑤 ∧ 𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))) → 𝑠 ⊆ (𝑤 × 𝑤))
20		df-ss 3883	. . . . . . . . . 10 ⊢ (𝑠 ⊆ (𝑤 × 𝑤) ↔ (𝑠 ∩ (𝑤 × 𝑤)) = 𝑠)
21	19, 20	sylib 221	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) ∧ (𝑤 ⊆ 𝑤 ∧ 𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))) → (𝑠 ∩ (𝑤 × 𝑤)) = 𝑠)
22	14, 21	eqtr2d 2778	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) ∧ (𝑤 ⊆ 𝑤 ∧ 𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))) → 𝑠 = 𝑡)
23	5	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) → 𝐴 ∈ 𝑉)
24		fpwwe2.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
25	24	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
26		simprl 771	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) → 𝑤𝑊𝑠)
27		simprr 773	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) → 𝑤𝑊𝑡)
28	1, 23, 25, 26, 27	fpwwe2lem9 10253	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) → ((𝑤 ⊆ 𝑤 ∧ 𝑠 = (𝑡 ∩ (𝑤 × 𝑤))) ∨ (𝑤 ⊆ 𝑤 ∧ 𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))))
29	13, 22, 28	mpjaodan 959	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) → 𝑠 = 𝑡)
30	29	ex 416	. . . . . 6 ⊢ (𝜑 → ((𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡) → 𝑠 = 𝑡))
31	30	alrimiv 1935	. . . . 5 ⊢ (𝜑 → ∀𝑡((𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡) → 𝑠 = 𝑡))
32	31	alrimivv 1936	. . . 4 ⊢ (𝜑 → ∀𝑤∀𝑠∀𝑡((𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡) → 𝑠 = 𝑡))
33		dffun2 6390	. . . 4 ⊢ (Fun 𝑊 ↔ (Rel 𝑊 ∧ ∀𝑤∀𝑠∀𝑡((𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡) → 𝑠 = 𝑡)))
34	3, 32, 33	sylanbrc 586	. . 3 ⊢ (𝜑 → Fun 𝑊)
35	34	funfnd 6411	. 2 ⊢ (𝜑 → 𝑊 Fn dom 𝑊)
36		vex 3412	. . . . 5 ⊢ 𝑠 ∈ V
37	36	elrn 5762	. . . 4 ⊢ (𝑠 ∈ ran 𝑊 ↔ ∃𝑤 𝑤𝑊𝑠)
38	2	releldmi 5817	. . . . . . . . . . . 12 ⊢ (𝑤𝑊𝑠 → 𝑤 ∈ dom 𝑊)
39	38	adantl 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤𝑊𝑠) → 𝑤 ∈ dom 𝑊)
40		elssuni 4851	. . . . . . . . . . 11 ⊢ (𝑤 ∈ dom 𝑊 → 𝑤 ⊆ ∪ dom 𝑊)
41	39, 40	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤𝑊𝑠) → 𝑤 ⊆ ∪ dom 𝑊)
42		fpwwe2.4	. . . . . . . . . 10 ⊢ 𝑋 = ∪ dom 𝑊
43	41, 42	sseqtrrdi 3952	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤𝑊𝑠) → 𝑤 ⊆ 𝑋)
44		xpss12 5566	. . . . . . . . 9 ⊢ ((𝑤 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑋) → (𝑤 × 𝑤) ⊆ (𝑋 × 𝑋))
45	43, 43, 44	syl2anc 587	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤𝑊𝑠) → (𝑤 × 𝑤) ⊆ (𝑋 × 𝑋))
46	17, 45	sstrd 3911	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤𝑊𝑠) → 𝑠 ⊆ (𝑋 × 𝑋))
47	46	ex 416	. . . . . 6 ⊢ (𝜑 → (𝑤𝑊𝑠 → 𝑠 ⊆ (𝑋 × 𝑋)))
48		velpw 4518	. . . . . 6 ⊢ (𝑠 ∈ 𝒫 (𝑋 × 𝑋) ↔ 𝑠 ⊆ (𝑋 × 𝑋))
49	47, 48	syl6ibr 255	. . . . 5 ⊢ (𝜑 → (𝑤𝑊𝑠 → 𝑠 ∈ 𝒫 (𝑋 × 𝑋)))
50	49	exlimdv 1941	. . . 4 ⊢ (𝜑 → (∃𝑤 𝑤𝑊𝑠 → 𝑠 ∈ 𝒫 (𝑋 × 𝑋)))
51	37, 50	syl5bi 245	. . 3 ⊢ (𝜑 → (𝑠 ∈ ran 𝑊 → 𝑠 ∈ 𝒫 (𝑋 × 𝑋)))
52	51	ssrdv 3907	. 2 ⊢ (𝜑 → ran 𝑊 ⊆ 𝒫 (𝑋 × 𝑋))
53		df-f 6384	. 2 ⊢ (𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋) ↔ (𝑊 Fn dom 𝑊 ∧ ran 𝑊 ⊆ 𝒫 (𝑋 × 𝑋)))
54	35, 52, 53	sylanbrc 586	1 ⊢ (𝜑 → 𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089  ∀wal 1541   = wceq 1543  ∃wex 1787   ∈ wcel 2110  ∀wral 3061  [wsbc 3694   ∩ cin 3865   ⊆ wss 3866  𝒫 cpw 4513  {csn 4541  ∪ cuni 4819   class class class wbr 5053  {copab 5115   We wwe 5508   × cxp 5549  ^◡ccnv 5550  dom cdm 5551  ran crn 5552   “ cima 5554  Rel wrel 5556  Fun wfun 6374   Fn wfn 6375  ⟶wf 6376  (class class class)co 7213

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 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-wrecs 8047  df-recs 8108  df-oi 9126

This theorem is referenced by:  fpwwe2lem12  10256  fpwwe2  10257