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Theorem marypha2lem2 8383
Description: Lemma for marypha2 8386. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)
Hypothesis
Ref Expression
marypha2lem.t 𝑇 = 𝑥𝐴 ({𝑥} × (𝐹𝑥))
Assertion
Ref Expression
marypha2lem2 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐹,𝑦
Allowed substitution hints:   𝑇(𝑥,𝑦)

Proof of Theorem marypha2lem2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 marypha2lem.t . 2 𝑇 = 𝑥𝐴 ({𝑥} × (𝐹𝑥))
2 sneq 4220 . . . 4 (𝑥 = 𝑧 → {𝑥} = {𝑧})
3 fveq2 6229 . . . 4 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
42, 3xpeq12d 5174 . . 3 (𝑥 = 𝑧 → ({𝑥} × (𝐹𝑥)) = ({𝑧} × (𝐹𝑧)))
54cbviunv 4591 . 2 𝑥𝐴 ({𝑥} × (𝐹𝑥)) = 𝑧𝐴 ({𝑧} × (𝐹𝑧))
6 df-xp 5149 . . . . 5 ({𝑧} × (𝐹𝑧)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))}
76a1i 11 . . . 4 (𝑧𝐴 → ({𝑧} × (𝐹𝑧)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))})
87iuneq2i 4571 . . 3 𝑧𝐴 ({𝑧} × (𝐹𝑧)) = 𝑧𝐴 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))}
9 iunopab 5041 . . 3 𝑧𝐴 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))}
10 velsn 4226 . . . . . . . 8 (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
11 equcom 1991 . . . . . . . 8 (𝑥 = 𝑧𝑧 = 𝑥)
1210, 11bitri 264 . . . . . . 7 (𝑥 ∈ {𝑧} ↔ 𝑧 = 𝑥)
1312anbi1i 731 . . . . . 6 ((𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧)) ↔ (𝑧 = 𝑥𝑦 ∈ (𝐹𝑧)))
1413rexbii 3070 . . . . 5 (∃𝑧𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧)) ↔ ∃𝑧𝐴 (𝑧 = 𝑥𝑦 ∈ (𝐹𝑧)))
15 fveq2 6229 . . . . . . 7 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
1615eleq2d 2716 . . . . . 6 (𝑧 = 𝑥 → (𝑦 ∈ (𝐹𝑧) ↔ 𝑦 ∈ (𝐹𝑥)))
1716ceqsrexbv 3368 . . . . 5 (∃𝑧𝐴 (𝑧 = 𝑥𝑦 ∈ (𝐹𝑧)) ↔ (𝑥𝐴𝑦 ∈ (𝐹𝑥)))
1814, 17bitri 264 . . . 4 (∃𝑧𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧)) ↔ (𝑥𝐴𝑦 ∈ (𝐹𝑥)))
1918opabbii 4750 . . 3 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
208, 9, 193eqtri 2677 . 2 𝑧𝐴 ({𝑧} × (𝐹𝑧)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
211, 5, 203eqtri 2677 1 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1523  wcel 2030  wrex 2942  {csn 4210   ciun 4552  {copab 4745   × cxp 5141  cfv 5926
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-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  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-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-xp 5149  df-iota 5889  df-fv 5934
This theorem is referenced by:  marypha2lem3  8384  marypha2lem4  8385  eulerpartlemgu  30567
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