Theorem fpwwe2cbv 10644

Description: Lemma for fpwwe2 10657. (Contributed by Mario Carneiro, 3-Jun-2015.)

Hypothesis

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))}

Assertion

Ref	Expression
fpwwe2cbv	⊢ 𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}

Distinct variable groups:   𝑦,𝑢   𝑟,𝑎,𝑠,𝑢,𝑣,𝑥,𝑦,𝑧,𝐹   𝐴,𝑎,𝑟,𝑠,𝑥,𝑧

Allowed substitution hints:   𝐴(𝑦,𝑣,𝑢)   𝑊(𝑥,𝑦,𝑧,𝑣,𝑢,𝑠,𝑟,𝑎)

Proof of Theorem fpwwe2cbv

Step	Hyp	Ref	Expression
1		fpwwe2.1	. 2 ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
2		simpl 482	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → 𝑥 = 𝑎)
3	2	sseq1d 3990	. . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑥 ⊆ 𝐴 ↔ 𝑎 ⊆ 𝐴))
4		simpr 484	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → 𝑟 = 𝑠)
5	2	sqxpeqd 5686	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑥 × 𝑥) = (𝑎 × 𝑎))
6	4, 5	sseq12d 3992	. . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑠 ⊆ (𝑎 × 𝑎)))
7	3, 6	anbi12d 632	. . . 4 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ↔ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎))))
8	4, 2	weeq12d 5643	. . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑟 We 𝑥 ↔ 𝑠 We 𝑎))
9		id 22	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑣 → 𝑢 = 𝑣)
10	9	sqxpeqd 5686	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (𝑢 × 𝑢) = (𝑣 × 𝑣))
11	10	ineq2d 4195	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑣 → (𝑟 ∩ (𝑢 × 𝑢)) = (𝑟 ∩ (𝑣 × 𝑣)))
12	9, 11	oveq12d 7423	. . . . . . . . . 10 ⊢ (𝑢 = 𝑣 → (𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = (𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))))
13	12	eqeq1d 2737	. . . . . . . . 9 ⊢ (𝑢 = 𝑣 → ((𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑦))
14	13	cbvsbcvw 3799	. . . . . . . 8 ⊢ ([(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ [(◡𝑟 “ {𝑦}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑦)
15		sneq 4611	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
16	15	imaeq2d 6047	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (◡𝑟 “ {𝑦}) = (◡𝑟 “ {𝑧}))
17		eqeq2 2747	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → ((𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑦 ↔ (𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧))
18	16, 17	sbceqbid 3772	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ([(◡𝑟 “ {𝑦}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑦 ↔ [(◡𝑟 “ {𝑧}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧))
19	14, 18	bitrid 283	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ([(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ [(◡𝑟 “ {𝑧}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧))
20	19	cbvralvw 3220	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ ∀𝑧 ∈ 𝑥 [(◡𝑟 “ {𝑧}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧)
21	4	cnveqd 5855	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → ◡𝑟 = ◡𝑠)
22	21	imaeq1d 6046	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (◡𝑟 “ {𝑧}) = (◡𝑠 “ {𝑧}))
23	4	ineq1d 4194	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑟 ∩ (𝑣 × 𝑣)) = (𝑠 ∩ (𝑣 × 𝑣)))
24	23	oveq2d 7421	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = (𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))))
25	24	eqeq1d 2737	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → ((𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧 ↔ (𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))
26	22, 25	sbceqbid 3772	. . . . . . 7 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → ([(◡𝑟 “ {𝑧}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧 ↔ [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))
27	2, 26	raleqbidv 3325	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (∀𝑧 ∈ 𝑥 [(◡𝑟 “ {𝑧}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧 ↔ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))
28	20, 27	bitrid 283	. . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))
29	8, 28	anbi12d 632	. . . 4 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → ((𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦) ↔ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧)))
30	7, 29	anbi12d 632	. . 3 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦)) ↔ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))))
31	30	cbvopabv 5192	. 2 ⊢ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))} = {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}
32	1, 31	eqtri 2758	1 ⊢ 𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))}

Syntax hints:   ∧ wa 395   = wceq 1540  ∀wral 3051  [wsbc 3765   ∩ cin 3925   ⊆ wss 3926  {csn 4601  {copab 5181   We wwe 5605   × cxp 5652  ^◡ccnv 5653   “ cima 5657  (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

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-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-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:  fpwwe2lem11  10655  fpwwe2lem12  10656  canthwe  10665  pwfseqlem5  10677