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Theorem marypha2lem2 9433
Description: Lemma for marypha2 9436. 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 4638 . . . 4 (𝑥 = 𝑧 → {𝑥} = {𝑧})
3 fveq2 6891 . . . 4 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
42, 3xpeq12d 5707 . . 3 (𝑥 = 𝑧 → ({𝑥} × (𝐹𝑥)) = ({𝑧} × (𝐹𝑧)))
54cbviunv 5043 . 2 𝑥𝐴 ({𝑥} × (𝐹𝑥)) = 𝑧𝐴 ({𝑧} × (𝐹𝑧))
6 df-xp 5682 . . . . 5 ({𝑧} × (𝐹𝑧)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))}
76a1i 11 . . . 4 (𝑧𝐴 → ({𝑧} × (𝐹𝑧)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))})
87iuneq2i 5018 . . 3 𝑧𝐴 ({𝑧} × (𝐹𝑧)) = 𝑧𝐴 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))}
9 iunopab 5559 . . 3 𝑧𝐴 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))}
10 velsn 4644 . . . . . . . 8 (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
11 equcom 2021 . . . . . . . 8 (𝑥 = 𝑧𝑧 = 𝑥)
1210, 11bitri 274 . . . . . . 7 (𝑥 ∈ {𝑧} ↔ 𝑧 = 𝑥)
1312anbi1i 624 . . . . . 6 ((𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧)) ↔ (𝑧 = 𝑥𝑦 ∈ (𝐹𝑧)))
1413rexbii 3094 . . . . 5 (∃𝑧𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧)) ↔ ∃𝑧𝐴 (𝑧 = 𝑥𝑦 ∈ (𝐹𝑧)))
15 fveq2 6891 . . . . . . 7 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
1615eleq2d 2819 . . . . . 6 (𝑧 = 𝑥 → (𝑦 ∈ (𝐹𝑧) ↔ 𝑦 ∈ (𝐹𝑥)))
1716ceqsrexbv 3644 . . . . 5 (∃𝑧𝐴 (𝑧 = 𝑥𝑦 ∈ (𝐹𝑧)) ↔ (𝑥𝐴𝑦 ∈ (𝐹𝑥)))
1814, 17bitri 274 . . . 4 (∃𝑧𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧)) ↔ (𝑥𝐴𝑦 ∈ (𝐹𝑥)))
1918opabbii 5215 . . 3 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
208, 9, 193eqtri 2764 . 2 𝑧𝐴 ({𝑧} × (𝐹𝑧)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
211, 5, 203eqtri 2764 1 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wcel 2106  wrex 3070  {csn 4628   ciun 4997  {copab 5210   × cxp 5674  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-xp 5682  df-iota 6495  df-fv 6551
This theorem is referenced by:  marypha2lem3  9434  marypha2lem4  9435  eulerpartlemgu  33445
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