Theorem fpwwe2cbv 9452

Description: Lemma for fpwwe2 9465. (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 473	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → 𝑥 = 𝑎)
3	2	sseq1d 3632	. . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑥 ⊆ 𝐴 ↔ 𝑎 ⊆ 𝐴))
4		simpr 477	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → 𝑟 = 𝑠)
5	2	sqxpeqd 5141	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑥 × 𝑥) = (𝑎 × 𝑎))
6	4, 5	sseq12d 3634	. . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑠 ⊆ (𝑎 × 𝑎)))
7	3, 6	anbi12d 747	. . . 4 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ↔ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎))))
8		weeq2 5103	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑟 We 𝑥 ↔ 𝑟 We 𝑎))
9		weeq1 5102	. . . . . 6 ⊢ (𝑟 = 𝑠 → (𝑟 We 𝑎 ↔ 𝑠 We 𝑎))
10	8, 9	sylan9bb 736	. . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑟 We 𝑥 ↔ 𝑠 We 𝑎))
11		id 22	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑣 → 𝑢 = 𝑣)
12	11	sqxpeqd 5141	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (𝑢 × 𝑢) = (𝑣 × 𝑣))
13	12	ineq2d 3814	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑣 → (𝑟 ∩ (𝑢 × 𝑢)) = (𝑟 ∩ (𝑣 × 𝑣)))
14	11, 13	oveq12d 6668	. . . . . . . . . 10 ⊢ (𝑢 = 𝑣 → (𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = (𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))))
15	14	eqeq1d 2624	. . . . . . . . 9 ⊢ (𝑢 = 𝑣 → ((𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑦))
16	15	cbvsbcv 3465	. . . . . . . 8 ⊢ ([(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ [(◡𝑟 “ {𝑦}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑦)
17		sneq 4187	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
18	17	imaeq2d 5466	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (◡𝑟 “ {𝑦}) = (◡𝑟 “ {𝑧}))
19		eqeq2 2633	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → ((𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑦 ↔ (𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧))
20	18, 19	sbceqbid 3442	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → ([(◡𝑟 “ {𝑦}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑦 ↔ [(◡𝑟 “ {𝑧}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧))
21	16, 20	syl5bb 272	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ([(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ [(◡𝑟 “ {𝑧}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧))
22	21	cbvralv 3171	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ ∀𝑧 ∈ 𝑥 [(◡𝑟 “ {𝑧}) / 𝑣](𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧)
23	4	cnveqd 5298	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → ◡𝑟 = ◡𝑠)
24	23	imaeq1d 5465	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (◡𝑟 “ {𝑧}) = (◡𝑠 “ {𝑧}))
25	4	ineq1d 3813	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑟 ∩ (𝑣 × 𝑣)) = (𝑠 ∩ (𝑣 × 𝑣)))
26	25	oveq2d 6666	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = (𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))))
27	26	eqeq1d 2624	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → ((𝑣𝐹(𝑟 ∩ (𝑣 × 𝑣))) = 𝑧 ↔ (𝑣𝐹(𝑠 ∩ (𝑣 × 𝑣))) = 𝑧))
28	24, 27	sbceqbid 3442	. . . . . . 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 3152	. . . . . 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 272	. . . . 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 747	. . . 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 747	. . 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 4722	. 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 2644	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 384   = wceq 1483  ∀wral 2912  [wsbc 3435   ∩ cin 3573   ⊆ wss 3574  {csn 4177  {copab 4712   We wwe 5072   × cxp 5112  ^◡ccnv 5113   “ cima 5117  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  fpwwe2lem12  9463  fpwwe2lem13  9464  canthwe  9473  pwfseqlem5  9485