Theorem fpwwe2lem2 10526

Description: Lemma for fpwwe2 10537. (Contributed by Mario Carneiro, 19-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)

Assertion

Ref	Expression
fpwwe2lem2	⊢ (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))

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   𝑊,𝑟,𝑢,𝑥,𝑦

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

Proof of Theorem fpwwe2lem2

Step	Hyp	Ref	Expression
1		fpwwe2.1	. . . . 5 ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
2	1	relopabiv 5763	. . . 4 ⊢ Rel 𝑊
3	2	a1i 11	. . 3 ⊢ (𝜑 → Rel 𝑊)
4		brrelex12 5671	. . 3 ⊢ ((Rel 𝑊 ∧ 𝑋𝑊𝑅) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
5	3, 4	sylan 580	. 2 ⊢ ((𝜑 ∧ 𝑋𝑊𝑅) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
6		fpwwe2.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → 𝐴 ∈ 𝑉)
8		simprll 778	. . . 4 ⊢ ((𝜑 ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → 𝑋 ⊆ 𝐴)
9	7, 8	ssexd 5263	. . 3 ⊢ ((𝜑 ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → 𝑋 ∈ V)
10	9, 9	xpexd 7687	. . . 4 ⊢ ((𝜑 ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → (𝑋 × 𝑋) ∈ V)
11		simprlr 779	. . . 4 ⊢ ((𝜑 ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → 𝑅 ⊆ (𝑋 × 𝑋))
12	10, 11	ssexd 5263	. . 3 ⊢ ((𝜑 ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → 𝑅 ∈ V)
13	9, 12	jca 511	. 2 ⊢ ((𝜑 ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
14		simpl 482	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → 𝑥 = 𝑋)
15	14	sseq1d 3967	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑥 ⊆ 𝐴 ↔ 𝑋 ⊆ 𝐴))
16		simpr 484	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
17	14	sqxpeqd 5651	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑥 × 𝑥) = (𝑋 × 𝑋))
18	16, 17	sseq12d 3969	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑅 ⊆ (𝑋 × 𝑋)))
19	15, 18	anbi12d 632	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ↔ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋))))
20	16, 14	weeq12d 5608	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑟 We 𝑥 ↔ 𝑅 We 𝑋))
21	16	cnveqd 5818	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → ◡𝑟 = ◡𝑅)
22	21	imaeq1d 6010	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (◡𝑟 “ {𝑦}) = (◡𝑅 “ {𝑦}))
23	16	ineq1d 4170	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑟 ∩ (𝑢 × 𝑢)) = (𝑅 ∩ (𝑢 × 𝑢)))
24	23	oveq2d 7365	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = (𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))))
25	24	eqeq1d 2731	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → ((𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))
26	22, 25	sbceqbid 3749	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → ([(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))
27	14, 26	raleqbidv 3309	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))
28	20, 27	anbi12d 632	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → ((𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦) ↔ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
29	19, 28	anbi12d 632	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦)) ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
30	29, 1	brabga 5477	. 2 ⊢ ((𝑋 ∈ V ∧ 𝑅 ∈ V) → (𝑋𝑊𝑅 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
31	5, 13, 30	pm5.21nd 801	1 ⊢ (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3436  [wsbc 3742   ∩ cin 3902   ⊆ wss 3903  {csn 4577   class class class wbr 5092  {copab 5154   We wwe 5571   × cxp 5617  ^◡ccnv 5618   “ cima 5622  Rel wrel 5624  (class class class)co 7349

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-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fv 6490  df-ov 7352

This theorem is referenced by:  fpwwe2lem3  10527  fpwwe2lem5  10529  fpwwe2lem6  10530  fpwwe2lem8  10532  fpwwe2lem10  10534  fpwwe2lem11  10535  fpwwe2lem12  10536  fpwwe2  10537  canthwelem  10544  pwfseqlem4  10556