Theorem fpwwe2lem2 10646

Description: Lemma for fpwwe2 10657. (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 5799	. . . 4 ⊢ Rel 𝑊
3	2	a1i 11	. . 3 ⊢ (𝜑 → Rel 𝑊)
4		brrelex12 5706	. . 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 5294	. . 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 7745	. . . 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 5294	. . 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 3990	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑥 ⊆ 𝐴 ↔ 𝑋 ⊆ 𝐴))
16		simpr 484	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
17	14	sqxpeqd 5686	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑥 × 𝑥) = (𝑋 × 𝑋))
18	16, 17	sseq12d 3992	. . . . 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 5643	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑟 We 𝑥 ↔ 𝑅 We 𝑋))
21	16	cnveqd 5855	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → ◡𝑟 = ◡𝑅)
22	21	imaeq1d 6046	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (◡𝑟 “ {𝑦}) = (◡𝑅 “ {𝑦}))
23	16	ineq1d 4194	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑟 ∩ (𝑢 × 𝑢)) = (𝑅 ∩ (𝑢 × 𝑢)))
24	23	oveq2d 7421	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = (𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))))
25	24	eqeq1d 2737	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → ((𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))
26	22, 25	sbceqbid 3772	. . . . . 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 3325	. . . . 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 5509	. 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 2108  ∀wral 3051  Vcvv 3459  [wsbc 3765   ∩ cin 3925   ⊆ wss 3926  {csn 4601   class class class wbr 5119  {copab 5181   We wwe 5605   × cxp 5652  ^◡ccnv 5653   “ cima 5657  Rel wrel 5659  (class class class)co 7405

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

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 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fv 6539  df-ov 7408

This theorem is referenced by:  fpwwe2lem3  10647  fpwwe2lem5  10649  fpwwe2lem6  10650  fpwwe2lem8  10652  fpwwe2lem10  10654  fpwwe2lem11  10655  fpwwe2lem12  10656  fpwwe2  10657  canthwelem  10664  pwfseqlem4  10676