MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fpwwe2lem5 Structured version   Visualization version   GIF version

Theorem fpwwe2lem5 9494
Description: Lemma for fpwwe2 9503. (Contributed by Mario Carneiro, 15-May-2015.)
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 (𝜑𝐴 ∈ V)
fpwwe2.3 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
Assertion
Ref Expression
fpwwe2lem5 ((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)

Proof of Theorem fpwwe2lem5
StepHypRef Expression
1 fpwwe2.2 . . . . 5 (𝜑𝐴 ∈ V)
21adantr 480 . . . 4 ((𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝐴 ∈ V)
3 simpr1 1087 . . . 4 ((𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑋𝐴)
42, 3ssexd 4838 . . 3 ((𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑋 ∈ V)
5 xpexg 7002 . . . . 5 ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑋 × 𝑋) ∈ V)
64, 4, 5syl2anc 694 . . . 4 ((𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋 × 𝑋) ∈ V)
7 simpr2 1088 . . . 4 ((𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑅 ⊆ (𝑋 × 𝑋))
86, 7ssexd 4838 . . 3 ((𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → 𝑅 ∈ V)
94, 8jca 553 . 2 ((𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋 ∈ V ∧ 𝑅 ∈ V))
10 sseq1 3659 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
11 xpeq12 5168 . . . . . . . 8 ((𝑥 = 𝑋𝑥 = 𝑋) → (𝑥 × 𝑥) = (𝑋 × 𝑋))
1211anidms 678 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 × 𝑥) = (𝑋 × 𝑋))
1312sseq2d 3666 . . . . . 6 (𝑥 = 𝑋 → (𝑟 ⊆ (𝑥 × 𝑥) ↔ 𝑟 ⊆ (𝑋 × 𝑋)))
14 weeq2 5132 . . . . . 6 (𝑥 = 𝑋 → (𝑟 We 𝑥𝑟 We 𝑋))
1510, 13, 143anbi123d 1439 . . . . 5 (𝑥 = 𝑋 → ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ↔ (𝑋𝐴𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋)))
1615anbi2d 740 . . . 4 (𝑥 = 𝑋 → ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) ↔ (𝜑 ∧ (𝑋𝐴𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋))))
17 oveq1 6697 . . . . 5 (𝑥 = 𝑋 → (𝑥𝐹𝑟) = (𝑋𝐹𝑟))
1817eleq1d 2715 . . . 4 (𝑥 = 𝑋 → ((𝑥𝐹𝑟) ∈ 𝐴 ↔ (𝑋𝐹𝑟) ∈ 𝐴))
1916, 18imbi12d 333 . . 3 (𝑥 = 𝑋 → (((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴) ↔ ((𝜑 ∧ (𝑋𝐴𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋)) → (𝑋𝐹𝑟) ∈ 𝐴)))
20 sseq1 3659 . . . . . 6 (𝑟 = 𝑅 → (𝑟 ⊆ (𝑋 × 𝑋) ↔ 𝑅 ⊆ (𝑋 × 𝑋)))
21 weeq1 5131 . . . . . 6 (𝑟 = 𝑅 → (𝑟 We 𝑋𝑅 We 𝑋))
2220, 213anbi23d 1442 . . . . 5 (𝑟 = 𝑅 → ((𝑋𝐴𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋) ↔ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)))
2322anbi2d 740 . . . 4 (𝑟 = 𝑅 → ((𝜑 ∧ (𝑋𝐴𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋)) ↔ (𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋))))
24 oveq2 6698 . . . . 5 (𝑟 = 𝑅 → (𝑋𝐹𝑟) = (𝑋𝐹𝑅))
2524eleq1d 2715 . . . 4 (𝑟 = 𝑅 → ((𝑋𝐹𝑟) ∈ 𝐴 ↔ (𝑋𝐹𝑅) ∈ 𝐴))
2623, 25imbi12d 333 . . 3 (𝑟 = 𝑅 → (((𝜑 ∧ (𝑋𝐴𝑟 ⊆ (𝑋 × 𝑋) ∧ 𝑟 We 𝑋)) → (𝑋𝐹𝑟) ∈ 𝐴) ↔ ((𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋𝐹𝑅) ∈ 𝐴)))
27 fpwwe2.3 . . 3 ((𝜑 ∧ (𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴)
2819, 26, 27vtocl2g 3301 . 2 ((𝑋 ∈ V ∧ 𝑅 ∈ V) → ((𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋𝐹𝑅) ∈ 𝐴))
299, 28mpcom 38 1 ((𝜑 ∧ (𝑋𝐴𝑅 ⊆ (𝑋 × 𝑋) ∧ 𝑅 We 𝑋)) → (𝑋𝐹𝑅) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  [wsbc 3468  cin 3606  wss 3607  {csn 4210  {copab 4745   We wwe 5101   × cxp 5141  ccnv 5142  cima 5146  (class class class)co 6690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-iota 5889  df-fv 5934  df-ov 6693
This theorem is referenced by:  fpwwe2lem13  9502
  Copyright terms: Public domain W3C validator