Theorem fpwwelem 10568

Description: Lemma for fpwwe 10569. (Contributed by Mario Carneiro, 15-May-2015.) (Revised by AV, 20-Jul-2024.)

Hypotheses

Ref	Expression
fpwwe.1	⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))}
fpwwe.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)

Assertion

Ref	Expression
fpwwelem	⊢ (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝐹‘(◡𝑅 “ {𝑦})) = 𝑦))))

Distinct variable groups:   𝑥,𝑟,𝐴   𝑦,𝑟,𝐹,𝑥   𝜑,𝑟,𝑥,𝑦   𝑅,𝑟,𝑥,𝑦   𝑋,𝑟,𝑥,𝑦   𝑊,𝑟,𝑥,𝑦

Allowed substitution hints:   𝐴(𝑦)   𝑉(𝑥,𝑦,𝑟)

Proof of Theorem fpwwelem

Step	Hyp	Ref	Expression
1		fpwwe.1	. . . . 5 ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦))}
2	1	relopabiv 5776	. . . 4 ⊢ Rel 𝑊
3	2	a1i 11	. . 3 ⊢ (𝜑 → Rel 𝑊)
4		brrelex12 5683	. . 3 ⊢ ((Rel 𝑊 ∧ 𝑋𝑊𝑅) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
5	3, 4	sylan 581	. 2 ⊢ ((𝜑 ∧ 𝑋𝑊𝑅) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
6		fpwwe.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝐹‘(◡𝑅 “ {𝑦})) = 𝑦))) → 𝐴 ∈ 𝑉)
8		simprll 779	. . . 4 ⊢ ((𝜑 ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝐹‘(◡𝑅 “ {𝑦})) = 𝑦))) → 𝑋 ⊆ 𝐴)
9	7, 8	ssexd 5265	. . 3 ⊢ ((𝜑 ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝐹‘(◡𝑅 “ {𝑦})) = 𝑦))) → 𝑋 ∈ V)
10	9, 9	xpexd 7705	. . . 4 ⊢ ((𝜑 ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝐹‘(◡𝑅 “ {𝑦})) = 𝑦))) → (𝑋 × 𝑋) ∈ V)
11		simprlr 780	. . . 4 ⊢ ((𝜑 ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝐹‘(◡𝑅 “ {𝑦})) = 𝑦))) → 𝑅 ⊆ (𝑋 × 𝑋))
12	10, 11	ssexd 5265	. . 3 ⊢ ((𝜑 ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝐹‘(◡𝑅 “ {𝑦})) = 𝑦))) → 𝑅 ∈ V)
13	9, 12	jca 511	. 2 ⊢ ((𝜑 ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝐹‘(◡𝑅 “ {𝑦})) = 𝑦))) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
14		simpl 482	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → 𝑥 = 𝑋)
15	14	sseq1d 3953	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑥 ⊆ 𝐴 ↔ 𝑋 ⊆ 𝐴))
16		simpr 484	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
17	14	sqxpeqd 5663	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑥 × 𝑥) = (𝑋 × 𝑋))
18	16, 17	sseq12d 3955	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑅 ⊆ (𝑋 × 𝑋)))
19	15, 18	anbi12d 633	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ↔ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋))))
20	16, 14	weeq12d 5620	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑟 We 𝑥 ↔ 𝑅 We 𝑋))
21	16	cnveqd 5830	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → ◡𝑟 = ◡𝑅)
22	21	imaeq1d 6024	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (◡𝑟 “ {𝑦}) = (◡𝑅 “ {𝑦}))
23	22	fveqeq2d 6848	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → ((𝐹‘(◡𝑟 “ {𝑦})) = 𝑦 ↔ (𝐹‘(◡𝑅 “ {𝑦})) = 𝑦))
24	14, 23	raleqbidv 3311	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦 ↔ ∀𝑦 ∈ 𝑋 (𝐹‘(◡𝑅 “ {𝑦})) = 𝑦))
25	20, 24	anbi12d 633	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → ((𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦) ↔ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝐹‘(◡𝑅 “ {𝑦})) = 𝑦)))
26	19, 25	anbi12d 633	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝐹‘(◡𝑟 “ {𝑦})) = 𝑦)) ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝐹‘(◡𝑅 “ {𝑦})) = 𝑦))))
27	26, 1	brabga 5489	. 2 ⊢ ((𝑋 ∈ V ∧ 𝑅 ∈ V) → (𝑋𝑊𝑅 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝐹‘(◡𝑅 “ {𝑦})) = 𝑦))))
28	5, 13, 27	pm5.21nd 802	1 ⊢ (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 (𝐹‘(◡𝑅 “ {𝑦})) = 𝑦))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  Vcvv 3429   ⊆ wss 3889  {csn 4567   class class class wbr 5085  {copab 5147   We wwe 5583   × cxp 5629  ^◡ccnv 5630   “ cima 5634  Rel wrel 5636  ‘cfv 6498

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fv 6506

This theorem is referenced by:  canth4  10570  canthnumlem  10571  canthp1lem2  10576