Theorem fpwwe2lem2 10710

Description: Lemma for fpwwe2 10721. (Contributed by Mario Carneiro, 19-May-2015.) (Revised by AV, 20-Jul-2024.)

Hypotheses

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))}
fpwwe2.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)

Assertion

Ref	Expression
fpwwe2lem2	⊢ (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))

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

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

Proof of Theorem fpwwe2lem2

Step	Hyp	Ref	Expression
1		fpwwe2.1	. . . . 5 ⊢ 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
2	1	relopabiv 5798	. . . 4 ⊢ Rel 𝑊
3	2	a1i 11	. . 3 ⊢ (𝜑 → Rel 𝑊)
4		brrelex12 5703	. . 3 ⊢ ((Rel 𝑊 ∧ 𝑋𝑊𝑅) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
5	3, 4	sylan 592	. 2 ⊢ ((𝜑 ∧ 𝑋𝑊𝑅) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
6		fpwwe2.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7	6	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → 𝐴 ∈ 𝑉)
8		simprll 791	. . . 4 ⊢ ((𝜑 ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → 𝑋 ⊆ 𝐴)
9	7, 8	ssexd 5286	. . 3 ⊢ ((𝜑 ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → 𝑋 ∈ V)
10	9, 9	xpexd 7763	. . . 4 ⊢ ((𝜑 ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → (𝑋 × 𝑋) ∈ V)
11		simprlr 792	. . . 4 ⊢ ((𝜑 ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → 𝑅 ⊆ (𝑋 × 𝑋))
12	10, 11	ssexd 5286	. . 3 ⊢ ((𝜑 ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → 𝑅 ∈ V)
13	9, 12	jca 521	. 2 ⊢ ((𝜑 ∧ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
14		simpl 488	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → 𝑥 = 𝑋)
15	14	sseq1d 3962	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑥 ⊆ 𝐴 ↔ 𝑋 ⊆ 𝐴))
16		simpr 490	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
17	14	sqxpeqd 5683	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑥 × 𝑥) = (𝑋 × 𝑋))
18	16, 17	sseq12d 3964	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑅 ⊆ (𝑋 × 𝑋)))
19	15, 18	anbi12d 644	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ↔ (𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋))))
20	16, 14	weeq12d 5640	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑟 We 𝑥 ↔ 𝑅 We 𝑋))
21	16	cnveqd 5853	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → ◡𝑟 = ◡𝑅)
22	21	imaeq1d 6051	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (◡𝑟 “ {𝑦}) = (◡𝑅 “ {𝑦}))
23	16	ineq1d 4165	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑟 ∩ (𝑢 × 𝑢)) = (𝑅 ∩ (𝑢 × 𝑢)))
24	23	oveq2d 7434	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = (𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))))
25	24	eqeq1d 2763	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → ((𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))
26	22, 25	sbceqbid 3746	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → ([(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))
27	14, 26	raleqbidv 3335	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))
28	20, 27	anbi12d 644	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → ((𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦) ↔ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
29	19, 28	anbi12d 644	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑟 = 𝑅) → (((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦)) ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
30	29, 1	brabga 5508	. 2 ⊢ ((𝑋 ∈ V ∧ 𝑅 ∈ V) → (𝑋𝑊𝑅 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
31	5, 13, 30	pm5.21nd 814	1 ⊢ (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451  [wsbc 3739   ∩ cin 3898   ⊆ wss 3899  {csn 4584   class class class wbr 5103  {copab 5167   We wwe 5603   × cxp 5649  ^◡ccnv 5650   “ cima 5654  Rel wrel 5656  (class class class)co 7418

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fv 6545  df-ov 7421

This theorem is used by:  fpwwe2lem3  10711  fpwwe2lem5  10713  fpwwe2lem6  10714  fpwwe2lem8  10716  fpwwe2lem10  10718  fpwwe2lem11  10719  fpwwe2lem12  10720  fpwwe2  10721  canthwelem  10728  pwfseqlem4  10740