Theorem fpwwe2cbv 10317

Description: Lemma for fpwwe2 10330. (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 3948	. . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑥 ⊆ 𝐴 ↔ 𝑎 ⊆ 𝐴))
4		simpr 484	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → 𝑟 = 𝑠)
5	2	sqxpeqd 5612	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑥 × 𝑥) = (𝑎 × 𝑎))
6	4, 5	sseq12d 3950	. . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑠 ⊆ (𝑎 × 𝑎)))
7	3, 6	anbi12d 630	. . . 4 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ↔ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎))))
8		weeq2 5569	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑟 We 𝑥 ↔ 𝑟 We 𝑎))
9		weeq1 5568	. . . . . 6 ⊢ (𝑟 = 𝑠 → (𝑟 We 𝑎 ↔ 𝑠 We 𝑎))
10	8, 9	sylan9bb 509	. . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑟 We 𝑥 ↔ 𝑠 We 𝑎))
11		id 22	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑣 → 𝑢 = 𝑣)
12	11	sqxpeqd 5612	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (𝑢 × 𝑢) = (𝑣 × 𝑣))
13	12	ineq2d 4143	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑣 → (𝑟 ∩ (𝑢 × 𝑢)) = (𝑟 ∩ (𝑣 × 𝑣)))
14	11, 13	oveq12d 7273	. . . . . . . . . 10 ⊢ (𝑢 = 𝑣 → (𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = (𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))))
15	14	eqeq1d 2740	. . . . . . . . 9 ⊢ (𝑢 = 𝑣 → ((𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑦))
16	15	cbvsbcvw 3746	. . . . . . . 8 ⊢ ([(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ [(◡𝑟 “ {𝑦}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑦)
17		sneq 4568	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
18	17	imaeq2d 5958	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (◡𝑟 “ {𝑦}) = (◡𝑟 “ {𝑧}))
19		eqeq2 2750	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → ((𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑦 ↔ (𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧))
20	18, 19	sbceqbid 3718	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ([(◡𝑟 “ {𝑦}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑦 ↔ [(◡𝑟 “ {𝑧}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧))
21	16, 20	syl5bb 282	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ([(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ [(◡𝑟 “ {𝑧}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧))
22	21	cbvralvw 3372	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ ∀𝑧 ∈ 𝑥 [(◡𝑟 “ {𝑧}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧)
23	4	cnveqd 5773	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → ◡𝑟 = ◡𝑠)
24	23	imaeq1d 5957	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (◡𝑟 “ {𝑧}) = (◡𝑠 “ {𝑧}))
25	4	ineq1d 4142	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑟 ∩ (𝑣 × 𝑣)) = (𝑠 ∩ (𝑣 × 𝑣)))
26	25	oveq2d 7271	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = (𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))))
27	26	eqeq1d 2740	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → ((𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧 ↔ (𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))
28	24, 27	sbceqbid 3718	. . . . . . 7 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → ([(◡𝑟 “ {𝑧}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧 ↔ [(◡𝑠 “ {𝑧}) / 𝑣](𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))
29	2, 28	raleqbidv 3327	. . . . . 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))
30	22, 29	syl5bb 282	. . . . 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))
31	10, 30	anbi12d 630	. . . 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)))
32	7, 31	anbi12d 630	. . 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))))
33	32	cbvopabv 5143	. 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))}
34	1, 33	eqtri 2766	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 1539  ∀wral 3063  [wsbc 3711   ∩ cin 3882   ⊆ wss 3883  {csn 4558  {copab 5132   We wwe 5534   × cxp 5578  ^◡ccnv 5579   “ cima 5583  (class class class)co 7255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fv 6426  df-ov 7258

This theorem is referenced by:  fpwwe2lem11  10328  fpwwe2lem12  10329  canthwe  10338  pwfseqlem5  10350