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Theorem fpwwe2lem2 10573
Description: Lemma for fpwwe2 10584. (Contributed by Mario Carneiro, 19-May-2015.) (Revised by AV, 20-Jul-2024.)
Hypotheses
Ref Expression
fpwwe2.1 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
fpwwe2.2 (𝜑𝐴𝑉)
Assertion
Ref Expression
fpwwe2lem2 (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(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
StepHypRef Expression
1 fpwwe2.1 . . . . 5 𝑊 = {⟨𝑥, 𝑟⟩ ∣ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))}
21relopabiv 5777 . . . 4 Rel 𝑊
32a1i 11 . . 3 (𝜑 → Rel 𝑊)
4 brrelex12 5685 . . 3 ((Rel 𝑊𝑋𝑊𝑅) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
53, 4sylan 581 . 2 ((𝜑𝑋𝑊𝑅) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
6 fpwwe2.2 . . . . 5 (𝜑𝐴𝑉)
76adantr 482 . . . 4 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → 𝐴𝑉)
8 simprll 778 . . . 4 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → 𝑋𝐴)
97, 8ssexd 5282 . . 3 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → 𝑋 ∈ V)
109, 9xpexd 7686 . . . 4 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → (𝑋 × 𝑋) ∈ V)
11 simprlr 779 . . . 4 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → 𝑅 ⊆ (𝑋 × 𝑋))
1210, 11ssexd 5282 . . 3 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → 𝑅 ∈ V)
139, 12jca 513 . 2 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
14 simpl 484 . . . . . 6 ((𝑥 = 𝑋𝑟 = 𝑅) → 𝑥 = 𝑋)
1514sseq1d 3976 . . . . 5 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑥𝐴𝑋𝐴))
16 simpr 486 . . . . . 6 ((𝑥 = 𝑋𝑟 = 𝑅) → 𝑟 = 𝑅)
1714sqxpeqd 5666 . . . . . 6 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑥 × 𝑥) = (𝑋 × 𝑋))
1816, 17sseq12d 3978 . . . . 5 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑅 ⊆ (𝑋 × 𝑋)))
1915, 18anbi12d 632 . . . 4 ((𝑥 = 𝑋𝑟 = 𝑅) → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ↔ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋))))
20 weeq2 5623 . . . . . 6 (𝑥 = 𝑋 → (𝑟 We 𝑥𝑟 We 𝑋))
21 weeq1 5622 . . . . . 6 (𝑟 = 𝑅 → (𝑟 We 𝑋𝑅 We 𝑋))
2220, 21sylan9bb 511 . . . . 5 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑟 We 𝑥𝑅 We 𝑋))
2316cnveqd 5832 . . . . . . . 8 ((𝑥 = 𝑋𝑟 = 𝑅) → 𝑟 = 𝑅)
2423imaeq1d 6013 . . . . . . 7 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑟 “ {𝑦}) = (𝑅 “ {𝑦}))
2516ineq1d 4172 . . . . . . . . 9 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑟 ∩ (𝑢 × 𝑢)) = (𝑅 ∩ (𝑢 × 𝑢)))
2625oveq2d 7374 . . . . . . . 8 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = (𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))))
2726eqeq1d 2735 . . . . . . 7 ((𝑥 = 𝑋𝑟 = 𝑅) → ((𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))
2824, 27sbceqbid 3747 . . . . . 6 ((𝑥 = 𝑋𝑟 = 𝑅) → ([(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦[(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))
2914, 28raleqbidv 3318 . . . . 5 ((𝑥 = 𝑋𝑟 = 𝑅) → (∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))
3022, 29anbi12d 632 . . . 4 ((𝑥 = 𝑋𝑟 = 𝑅) → ((𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦) ↔ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))
3119, 30anbi12d 632 . . 3 ((𝑥 = 𝑋𝑟 = 𝑅) → (((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦)) ↔ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
3231, 1brabga 5492 . 2 ((𝑋 ∈ V ∧ 𝑅 ∈ V) → (𝑋𝑊𝑅 ↔ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
335, 13, 32pm5.21nd 801 1 (𝜑 → (𝑋𝑊𝑅 ↔ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3061  Vcvv 3444  [wsbc 3740  cin 3910  wss 3911  {csn 4587   class class class wbr 5106  {copab 5168   We wwe 5588   × cxp 5632  ccnv 5633  cima 5637  Rel wrel 5639  (class class class)co 7358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fv 6505  df-ov 7361
This theorem is referenced by:  fpwwe2lem3  10574  fpwwe2lem5  10576  fpwwe2lem6  10577  fpwwe2lem8  10579  fpwwe2lem10  10581  fpwwe2lem11  10582  fpwwe2lem12  10583  fpwwe2  10584  canthwelem  10591  pwfseqlem4  10603
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