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Theorem fpwwe2lem2 10657
Description: Lemma for fpwwe2 10668. (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 5822 . . . 4 Rel 𝑊
32a1i 11 . . 3 (𝜑 → Rel 𝑊)
4 brrelex12 5730 . . 3 ((Rel 𝑊𝑋𝑊𝑅) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
53, 4sylan 578 . 2 ((𝜑𝑋𝑊𝑅) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
6 fpwwe2.2 . . . . 5 (𝜑𝐴𝑉)
76adantr 479 . . . 4 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → 𝐴𝑉)
8 simprll 777 . . . 4 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → 𝑋𝐴)
97, 8ssexd 5325 . . 3 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → 𝑋 ∈ V)
109, 9xpexd 7754 . . . 4 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → (𝑋 × 𝑋) ∈ V)
11 simprlr 778 . . . 4 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → 𝑅 ⊆ (𝑋 × 𝑋))
1210, 11ssexd 5325 . . 3 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → 𝑅 ∈ V)
139, 12jca 510 . 2 ((𝜑 ∧ ((𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋)) ∧ (𝑅 We 𝑋 ∧ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
14 simpl 481 . . . . . 6 ((𝑥 = 𝑋𝑟 = 𝑅) → 𝑥 = 𝑋)
1514sseq1d 4008 . . . . 5 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑥𝐴𝑋𝐴))
16 simpr 483 . . . . . 6 ((𝑥 = 𝑋𝑟 = 𝑅) → 𝑟 = 𝑅)
1714sqxpeqd 5710 . . . . . 6 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑥 × 𝑥) = (𝑋 × 𝑋))
1816, 17sseq12d 4010 . . . . 5 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑅 ⊆ (𝑋 × 𝑋)))
1915, 18anbi12d 630 . . . 4 ((𝑥 = 𝑋𝑟 = 𝑅) → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥)) ↔ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋))))
20 weeq2 5667 . . . . . 6 (𝑥 = 𝑋 → (𝑟 We 𝑥𝑟 We 𝑋))
21 weeq1 5666 . . . . . 6 (𝑟 = 𝑅 → (𝑟 We 𝑋𝑅 We 𝑋))
2220, 21sylan9bb 508 . . . . 5 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑟 We 𝑥𝑅 We 𝑋))
2316cnveqd 5878 . . . . . . . 8 ((𝑥 = 𝑋𝑟 = 𝑅) → 𝑟 = 𝑅)
2423imaeq1d 6063 . . . . . . 7 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑟 “ {𝑦}) = (𝑅 “ {𝑦}))
2516ineq1d 4209 . . . . . . . . 9 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑟 ∩ (𝑢 × 𝑢)) = (𝑅 ∩ (𝑢 × 𝑢)))
2625oveq2d 7435 . . . . . . . 8 ((𝑥 = 𝑋𝑟 = 𝑅) → (𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = (𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))))
2726eqeq1d 2727 . . . . . . 7 ((𝑥 = 𝑋𝑟 = 𝑅) → ((𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ (𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))
2824, 27sbceqbid 3780 . . . . . 6 ((𝑥 = 𝑋𝑟 = 𝑅) → ([(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦[(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))
2914, 28raleqbidv 3329 . . . . 5 ((𝑥 = 𝑋𝑟 = 𝑅) → (∀𝑦𝑥 [(𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦 ↔ ∀𝑦𝑋 [(𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))
3022, 29anbi12d 630 . . . 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 630 . . 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 5536 . 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 800 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 394   = wceq 1533  wcel 2098  wral 3050  Vcvv 3461  [wsbc 3773  cin 3943  wss 3944  {csn 4630   class class class wbr 5149  {copab 5211   We wwe 5632   × cxp 5676  ccnv 5677  cima 5681  Rel wrel 5683  (class class class)co 7419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fv 6557  df-ov 7422
This theorem is referenced by:  fpwwe2lem3  10658  fpwwe2lem5  10660  fpwwe2lem6  10661  fpwwe2lem8  10663  fpwwe2lem10  10665  fpwwe2lem11  10666  fpwwe2lem12  10667  fpwwe2  10668  canthwelem  10675  pwfseqlem4  10687
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