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Theorem fpwwe2lem4 10578
Description: Lemma for fpwwe2 10587. (Contributed by Mario Carneiro, 15-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)
fpwwe2.3 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
Assertion
Ref Expression
fpwwe2lem4 ((𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋𝐹𝑅) ∈ 𝐴)
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 fpwwe2lem4
StepHypRef Expression
1 fpwwe2.2 . . . . 5 (𝜑𝐴𝑉)
21adantr 482 . . . 4 ((𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝐴𝑉)
3 simpr1 1195 . . . 4 ((𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑋𝐴)
42, 3ssexd 5285 . . 3 ((𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑋 ∈ V)
54, 4xpexd 7689 . . . 4 ((𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋 × 𝑋) ∈ V)
6 simpr2 1196 . . . 4 ((𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑅 ⊆ (𝑋 × 𝑋))
75, 6ssexd 5285 . . 3 ((𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑅 ∈ V)
84, 7jca 513 . 2 ((𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
9 sseq1 3973 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
10 xpeq12 5662 . . . . . . . 8 ((𝑥 = 𝑋𝑥 = 𝑋) → (𝑥 × 𝑥) = (𝑋 × 𝑋))
1110anidms 568 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 × 𝑥) = (𝑋 × 𝑋))
1211sseq2d 3980 . . . . . 6 (𝑥 = 𝑋 → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑟 ⊆ (𝑋 × 𝑋)))
13 weeq2 5626 . . . . . 6 (𝑥 = 𝑋 → (𝑟 We 𝑥𝑟 We 𝑋))
149, 12, 133anbi123d 1437 . . . . 5 (𝑥 = 𝑋 → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝑋𝐴𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋)))
1514anbi2d 630 . . . 4 (𝑥 = 𝑋 → ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) ↔ (𝜑 ∧ (𝑋𝐴𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋))))
16 oveq1 7368 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐹𝑟) = (𝑋𝐹𝑟))
1716eleq1d 2819 . . . 4 (𝑥 = 𝑋 → ((𝑥𝐹𝑟) ∈ 𝐴 ↔ (𝑋𝐹𝑟) ∈ 𝐴))
1815, 17imbi12d 345 . . 3 (𝑥 = 𝑋 → (((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴) ↔ ((𝜑 ∧ (𝑋𝐴𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋)) → (𝑋𝐹𝑟) ∈ 𝐴)))
19 sseq1 3973 . . . . . 6 (𝑟 = 𝑅 → (𝑟 ⊆ (𝑋 × 𝑋) ↔ 𝑅 ⊆ (𝑋 × 𝑋)))
20 weeq1 5625 . . . . . 6 (𝑟 = 𝑅 → (𝑟 We 𝑋𝑅 We 𝑋))
2119, 203anbi23d 1440 . . . . 5 (𝑟 = 𝑅 → ((𝑋𝐴𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋) ↔ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)))
2221anbi2d 630 . . . 4 (𝑟 = 𝑅 → ((𝜑 ∧ (𝑋𝐴𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋)) ↔ (𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋))))
23 oveq2 7369 . . . . 5 (𝑟 = 𝑅 → (𝑋𝐹𝑟) = (𝑋𝐹𝑅))
2423eleq1d 2819 . . . 4 (𝑟 = 𝑅 → ((𝑋𝐹𝑟) ∈ 𝐴 ↔ (𝑋𝐹𝑅) ∈ 𝐴))
2522, 24imbi12d 345 . . 3 (𝑟 = 𝑅 → (((𝜑 ∧ (𝑋𝐴𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋)) → (𝑋𝐹𝑟) ∈ 𝐴) ↔ ((𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋𝐹𝑅) ∈ 𝐴)))
26 fpwwe2.3 . . 3 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
2718, 25, 26vtocl2g 3533 . 2 ((𝑋 ∈ V ∧ 𝑅 ∈ V) → ((𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋𝐹𝑅) ∈ 𝐴))
288, 27mpcom 38 1 ((𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋𝐹𝑅) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3061  Vcvv 3447  [wsbc 3743  cin 3913  wss 3914  {csn 4590  {copab 5171   We wwe 5591   × cxp 5635  ccnv 5636  cima 5640  (class class class)co 7361
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 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-iota 6452  df-fv 6508  df-ov 7364
This theorem is referenced by:  fpwwe2lem12  10586
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