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Theorem marypha2lem2 9379
Description: Lemma for marypha2 9382. 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 4601 . . . 4 (𝑥 = 𝑧 → {𝑥} = {𝑧})
3 fveq2 6847 . . . 4 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
42, 3xpeq12d 5669 . . 3 (𝑥 = 𝑧 → ({𝑥} × (𝐹𝑥)) = ({𝑧} × (𝐹𝑧)))
54cbviunv 5005 . 2 𝑥𝐴 ({𝑥} × (𝐹𝑥)) = 𝑧𝐴 ({𝑧} × (𝐹𝑧))
6 df-xp 5644 . . . . 5 ({𝑧} × (𝐹𝑧)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))}
76a1i 11 . . . 4 (𝑧𝐴 → ({𝑧} × (𝐹𝑧)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))})
87iuneq2i 4980 . . 3 𝑧𝐴 ({𝑧} × (𝐹𝑧)) = 𝑧𝐴 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))}
9 iunopab 5521 . . 3 𝑧𝐴 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))}
10 velsn 4607 . . . . . . . 8 (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
11 equcom 2022 . . . . . . . 8 (𝑥 = 𝑧𝑧 = 𝑥)
1210, 11bitri 275 . . . . . . 7 (𝑥 ∈ {𝑧} ↔ 𝑧 = 𝑥)
1312anbi1i 625 . . . . . 6 ((𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧)) ↔ (𝑧 = 𝑥𝑦 ∈ (𝐹𝑧)))
1413rexbii 3098 . . . . 5 (∃𝑧𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧)) ↔ ∃𝑧𝐴 (𝑧 = 𝑥𝑦 ∈ (𝐹𝑧)))
15 fveq2 6847 . . . . . . 7 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
1615eleq2d 2824 . . . . . 6 (𝑧 = 𝑥 → (𝑦 ∈ (𝐹𝑧) ↔ 𝑦 ∈ (𝐹𝑥)))
1716ceqsrexbv 3611 . . . . 5 (∃𝑧𝐴 (𝑧 = 𝑥𝑦 ∈ (𝐹𝑧)) ↔ (𝑥𝐴𝑦 ∈ (𝐹𝑥)))
1814, 17bitri 275 . . . 4 (∃𝑧𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧)) ↔ (𝑥𝐴𝑦 ∈ (𝐹𝑥)))
1918opabbii 5177 . . 3 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
208, 9, 193eqtri 2769 . 2 𝑧𝐴 ({𝑧} × (𝐹𝑧)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
211, 5, 203eqtri 2769 1 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wcel 2107  wrex 3074  {csn 4591   ciun 4959  {copab 5172   × cxp 5636  cfv 6501
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-11 2155  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-xp 5644  df-iota 6453  df-fv 6509
This theorem is referenced by:  marypha2lem3  9380  marypha2lem4  9381  eulerpartlemgu  33017
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