Theorem fpwwecbv 10055

Description: Lemma for fpwwe 10057. (Contributed by Mario Carneiro, 15-May-2015.)

Hypothesis

Ref	Expression
fpwwe.1	⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))}

Assertion

Ref	Expression
fpwwecbv	⊢ 𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 (𝐹‘(◡𝑠 “ {𝑧})) = 𝑧))}

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

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

Proof of Theorem fpwwecbv

Step	Hyp	Ref	Expression
1		fpwwe.1	. 2 ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))}
2		simpl 486	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → 𝑥 = 𝑎)
3	2	sseq1d 3946	. . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑥 ⊆ 𝐴 ↔ 𝑎 ⊆ 𝐴))
4		simpr 488	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → 𝑟 = 𝑠)
5	2	sqxpeqd 5551	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑥 × 𝑥) = (𝑎 × 𝑎))
6	4, 5	sseq12d 3948	. . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑠 ⊆ (𝑎 × 𝑎)))
7	3, 6	anbi12d 633	. . . 4 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ↔ (𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎))))
8		weeq2 5508	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑟 We 𝑥 ↔ 𝑟 We 𝑎))
9		weeq1 5507	. . . . . 6 ⊢ (𝑟 = 𝑠 → (𝑟 We 𝑎 ↔ 𝑠 We 𝑎))
10	8, 9	sylan9bb 513	. . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (𝑟 We 𝑥 ↔ 𝑠 We 𝑎))
11		sneq 4535	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
12	11	imaeq2d 5896	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (◡𝑟 “ {𝑦}) = (◡𝑟 “ {𝑧}))
13	12	fveq2d 6649	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝐹‘(◡𝑟 “ {𝑦})) = (𝐹‘(◡𝑟 “ {𝑧})))
14		id 22	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
15	13, 14	eqeq12d 2814	. . . . . . 7 ⊢ (𝑦 = 𝑧 → ((𝐹‘(◡𝑟 “ {𝑦})) = 𝑦 ↔ (𝐹‘(◡𝑟 “ {𝑧})) = 𝑧))
16	15	cbvralvw 3396	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦 ↔ ∀𝑧 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑧})) = 𝑧)
17	4	cnveqd 5710	. . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → ◡𝑟 = ◡𝑠)
18	17	imaeq1d 5895	. . . . . . . 8 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (◡𝑟 “ {𝑧}) = (◡𝑠 “ {𝑧}))
19	18	fveqeq2d 6653	. . . . . . 7 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → ((𝐹‘(◡𝑟 “ {𝑧})) = 𝑧 ↔ (𝐹‘(◡𝑠 “ {𝑧})) = 𝑧))
20	2, 19	raleqbidv 3354	. . . . . 6 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (∀𝑧 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑧})) = 𝑧 ↔ ∀𝑧 ∈ 𝑎 (𝐹‘(◡𝑠 “ {𝑧})) = 𝑧))
21	16, 20	syl5bb 286	. . . . 5 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦 ↔ ∀𝑧 ∈ 𝑎 (𝐹‘(◡𝑠 “ {𝑧})) = 𝑧))
22	10, 21	anbi12d 633	. . . 4 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → ((𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦) ↔ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 (𝐹‘(◡𝑠 “ {𝑧})) = 𝑧)))
23	7, 22	anbi12d 633	. . 3 ⊢ ((𝑥 = 𝑎 ∧ 𝑟 = 𝑠) → (((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦)) ↔ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 (𝐹‘(◡𝑠 “ {𝑧})) = 𝑧))))
24	23	cbvopabv 5102	. 2 ⊢ {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))} = {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 (𝐹‘(◡𝑠 “ {𝑧})) = 𝑧))}
25	1, 24	eqtri 2821	1 ⊢ 𝑊 = {⟨𝑎, 𝑠⟩ ∣ ((𝑎 ⊆ 𝐴 ∧ 𝑠 ⊆ (𝑎 × 𝑎)) ∧ (𝑠 We 𝑎 ∧ ∀𝑧 ∈ 𝑎 (𝐹‘(◡𝑠 “ {𝑧})) = 𝑧))}

Syntax hints:   ∧ wa 399   = wceq 1538  ∀wral 3106   ⊆ wss 3881  {csn 4525  {copab 5092   We wwe 5477   × cxp 5517  ^◡ccnv 5518   “ cima 5522  ‘cfv 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fv 6332

This theorem is referenced by:  canthnum  10060  canthp1  10065