Theorem fpwwe2cbv 10614

Description: Lemma for fpwwe2 10627. (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 487	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → 𝑥 = 𝑎)
3	2	sseq1d 3976	. . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑥 ⊆ 𝐴 ↔ 𝑎 ⊆ 𝐴))
4		simpr 489	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → 𝑟 = 𝑠)
5	2	sqxpeqd 5694	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑥 × 𝑥) = (𝑎 × 𝑎))
6	4, 5	sseq12d 3978	. . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑠 ⊆ (𝑎 × 𝑎)))
7	3, 6	anbi12d 643	. . . 4 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ↔ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎))))
8	4, 2	weeq12d 5651	. . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑟 We 𝑥 ↔ 𝑠 We 𝑎))
9		id 23	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑣 → 𝑢 = 𝑣)
10	9	sqxpeqd 5694	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (𝑢 × 𝑢) = (𝑣 × 𝑣))
11	10	ineq2d 4181	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑣 → (𝑟 ∩ (𝑢 × 𝑢)) = (𝑟 ∩ (𝑣 × 𝑣)))
12	9, 11	oveq12d 7429	. . . . . . . . . 10 ⊢ (𝑢 = 𝑣 → (𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = (𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))))
13	12	eqeq1d 2771	. . . . . . . . 9 ⊢ (𝑢 = 𝑣 → ((𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑦))
14	13	cbvsbcvw 3787	. . . . . . . 8 ⊢ ([(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ [(◡𝑟 “ {𝑦}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑦)
15		sneq 4604	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
16	15	imaeq2d 6063	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (◡𝑟 “ {𝑦}) = (◡𝑟 “ {𝑧}))
17		eqeq2 2781	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → ((𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑦 ↔ (𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧))
18	16, 17	sbceqbid 3760	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ([(◡𝑟 “ {𝑦}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑦 ↔ [(◡𝑟 “ {𝑧}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧))
19	14, 18	bitrid 286	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ([(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ [(◡𝑟 “ {𝑧}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧))
20	19	cbvralvw 3249	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ ∀𝑧 ∈ 𝑥 [(◡𝑟 “ {𝑧}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧)
21	4	cnveqd 5862	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → ◡𝑟 = ◡𝑠)
22	21	imaeq1d 6062	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (◡𝑟 “ {𝑧}) = (◡𝑠 “ {𝑧}))
23	4	ineq1d 4180	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑟 ∩ (𝑣 × 𝑣)) = (𝑠 ∩ (𝑣 × 𝑣)))
24	23	oveq2d 7427	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = (𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))))
25	24	eqeq1d 2771	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → ((𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧 ↔ (𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))
26	22, 25	sbceqbid 3760	. . . . . . 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 3345	. . . . . 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 286	. . . . 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 643	. . . 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 643	. . 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 5188	. 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 2792	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 400   = wceq 1567  ∀wral 3085  [wsbc 3753   ∩ cin 3912   ⊆ wss 3913  {csn 4594  {copab 5177   We wwe 5614   × cxp 5660  ^◡ccnv 5661   “ cima 5665  (class class class)co 7411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fv 6545  df-ov 7414

This theorem is referenced by:  fpwwe2lem11  10625  fpwwe2lem12  10626  canthwe  10635  pwfseqlem5  10647