Theorem fpwwe2lem10 10569

Description: Lemma for fpwwe2 10572. (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 5774	. . . . 5 ⊢ Rel 𝑊
3	2	a1i 11	. . . 4 ⊢ (𝜑 → Rel 𝑊)
4		simprr 772	. . . . . . . . 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 10561	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑤𝑊𝑡 ↔ ((𝑤 ⊆ 𝐴 ∧ 𝑡 ⊆ (𝑤 × 𝑤)) ∧ (𝑡 We 𝑤 ∧ ∀𝑦 ∈ 𝑤 [(◡𝑡 “ {𝑦}) / 𝑢](𝑢𝐹(𝑡 ∩ (𝑢 × 𝑢))) = 𝑦))))
7	6	simprbda 498	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤𝑊𝑡) → (𝑤 ⊆ 𝐴 ∧ 𝑡 ⊆ (𝑤 × 𝑤)))
8	7	simprd 495	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤𝑊𝑡) → 𝑡 ⊆ (𝑤 × 𝑤))
9	8	adantrl 716	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) → 𝑡 ⊆ (𝑤 × 𝑤))
10	9	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) ∧ (𝑤 ⊆ 𝑤 ∧ 𝑠 = (𝑡 ∩ (𝑤 × 𝑤)))) → 𝑡 ⊆ (𝑤 × 𝑤))
11		dfss2 3929	. . . . . . . . . 10 ⊢ (𝑡 ⊆ (𝑤 × 𝑤) ↔ (𝑡 ∩ (𝑤 × 𝑤)) = 𝑡)
12	10, 11	sylib 218	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) ∧ (𝑤 ⊆ 𝑤 ∧ 𝑠 = (𝑡 ∩ (𝑤 × 𝑤)))) → (𝑡 ∩ (𝑤 × 𝑤)) = 𝑡)
13	4, 12	eqtrd 2764	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) ∧ (𝑤 ⊆ 𝑤 ∧ 𝑠 = (𝑡 ∩ (𝑤 × 𝑤)))) → 𝑠 = 𝑡)
14		simprr 772	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) ∧ (𝑤 ⊆ 𝑤 ∧ 𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))) → 𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))
15	1, 5	fpwwe2lem2 10561	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑤𝑊𝑠 ↔ ((𝑤 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑤 × 𝑤)) ∧ (𝑠 We 𝑤 ∧ ∀𝑦 ∈ 𝑤 [(◡𝑠 “ {𝑦}) / 𝑢](𝑢𝐹(𝑠 ∩ (𝑢 × 𝑢))) = 𝑦))))
16	15	simprbda 498	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤𝑊𝑠) → (𝑤 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑤 × 𝑤)))
17	16	simprd 495	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤𝑊𝑠) → 𝑠 ⊆ (𝑤 × 𝑤))
18	17	adantrr 717	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) → 𝑠 ⊆ (𝑤 × 𝑤))
19	18	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) ∧ (𝑤 ⊆ 𝑤 ∧ 𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))) → 𝑠 ⊆ (𝑤 × 𝑤))
20		dfss2 3929	. . . . . . . . . 10 ⊢ (𝑠 ⊆ (𝑤 × 𝑤) ↔ (𝑠 ∩ (𝑤 × 𝑤)) = 𝑠)
21	19, 20	sylib 218	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) ∧ (𝑤 ⊆ 𝑤 ∧ 𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))) → (𝑠 ∩ (𝑤 × 𝑤)) = 𝑠)
22	14, 21	eqtr2d 2765	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) ∧ (𝑤 ⊆ 𝑤 ∧ 𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))) → 𝑠 = 𝑡)
23	5	adantr 480	. . . . . . . . 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 770	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) → 𝑤𝑊𝑠)
27		simprr 772	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) → 𝑤𝑊𝑡)
28	1, 23, 25, 26, 27	fpwwe2lem9 10568	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) → ((𝑤 ⊆ 𝑤 ∧ 𝑠 = (𝑡 ∩ (𝑤 × 𝑤))) ∨ (𝑤 ⊆ 𝑤 ∧ 𝑡 = (𝑠 ∩ (𝑤 × 𝑤)))))
29	13, 22, 28	mpjaodan 960	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡)) → 𝑠 = 𝑡)
30	29	ex 412	. . . . . 6 ⊢ (𝜑 → ((𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡) → 𝑠 = 𝑡))
31	30	alrimiv 1927	. . . . 5 ⊢ (𝜑 → ∀𝑡((𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡) → 𝑠 = 𝑡))
32	31	alrimivv 1928	. . . 4 ⊢ (𝜑 → ∀𝑤∀𝑠∀𝑡((𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡) → 𝑠 = 𝑡))
33		dffun2 6509	. . . 4 ⊢ (Fun 𝑊 ↔ (Rel 𝑊 ∧ ∀𝑤∀𝑠∀𝑡((𝑤𝑊𝑠 ∧ 𝑤𝑊𝑡) → 𝑠 = 𝑡)))
34	3, 32, 33	sylanbrc 583	. . 3 ⊢ (𝜑 → Fun 𝑊)
35	34	funfnd 6531	. 2 ⊢ (𝜑 → 𝑊 Fn dom 𝑊)
36		vex 3448	. . . . 5 ⊢ 𝑠 ∈ V
37	36	elrn 5847	. . . 4 ⊢ (𝑠 ∈ ran 𝑊 ↔ ∃𝑤 𝑤𝑊𝑠)
38	2	releldmi 5901	. . . . . . . . . . . 12 ⊢ (𝑤𝑊𝑠 → 𝑤 ∈ dom 𝑊)
39	38	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤𝑊𝑠) → 𝑤 ∈ dom 𝑊)
40		elssuni 4897	. . . . . . . . . . 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 3985	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤𝑊𝑠) → 𝑤 ⊆ 𝑋)
44		xpss12 5646	. . . . . . . . 9 ⊢ ((𝑤 ⊆ 𝑋 ∧ 𝑤 ⊆ 𝑋) → (𝑤 × 𝑤) ⊆ (𝑋 × 𝑋))
45	43, 43, 44	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑤𝑊𝑠) → (𝑤 × 𝑤) ⊆ (𝑋 × 𝑋))
46	17, 45	sstrd 3954	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤𝑊𝑠) → 𝑠 ⊆ (𝑋 × 𝑋))
47	46	ex 412	. . . . . 6 ⊢ (𝜑 → (𝑤𝑊𝑠 → 𝑠 ⊆ (𝑋 × 𝑋)))
48		velpw 4564	. . . . . 6 ⊢ (𝑠 ∈ 𝒫 (𝑋 × 𝑋) ↔ 𝑠 ⊆ (𝑋 × 𝑋))
49	47, 48	imbitrrdi 252	. . . . 5 ⊢ (𝜑 → (𝑤𝑊𝑠 → 𝑠 ∈ 𝒫 (𝑋 × 𝑋)))
50	49	exlimdv 1933	. . . 4 ⊢ (𝜑 → (∃𝑤 𝑤𝑊𝑠 → 𝑠 ∈ 𝒫 (𝑋 × 𝑋)))
51	37, 50	biimtrid 242	. . 3 ⊢ (𝜑 → (𝑠 ∈ ran 𝑊 → 𝑠 ∈ 𝒫 (𝑋 × 𝑋)))
52	51	ssrdv 3949	. 2 ⊢ (𝜑 → ran 𝑊 ⊆ 𝒫 (𝑋 × 𝑋))
53		df-f 6503	. 2 ⊢ (𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋) ↔ (𝑊 Fn dom 𝑊 ∧ ran 𝑊 ⊆ 𝒫 (𝑋 × 𝑋)))
54	35, 52, 53	sylanbrc 583	1 ⊢ (𝜑 → 𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  [wsbc 3750   ∩ cin 3910   ⊆ wss 3911  𝒫 cpw 4559  {csn 4585  ∪ cuni 4867   class class class wbr 5102  {copab 5164   We wwe 5583   × cxp 5629  ^◡ccnv 5630  dom cdm 5631  ran crn 5632   “ cima 5634  Rel wrel 5636  Fun wfun 6493   Fn wfn 6494  ⟶wf 6495  (class class class)co 7369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-oi 9439

This theorem is referenced by:  fpwwe2lem12  10571  fpwwe2  10572