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Theorem marypha2lem2 8202
Description: Lemma for marypha2 8205. 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 4134 . . . 4 (𝑥 = 𝑧 → {𝑥} = {𝑧})
3 fveq2 6087 . . . 4 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
42, 3xpeq12d 5053 . . 3 (𝑥 = 𝑧 → ({𝑥} × (𝐹𝑥)) = ({𝑧} × (𝐹𝑧)))
54cbviunv 4489 . 2 𝑥𝐴 ({𝑥} × (𝐹𝑥)) = 𝑧𝐴 ({𝑧} × (𝐹𝑧))
6 df-xp 5033 . . . . 5 ({𝑧} × (𝐹𝑧)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))}
76a1i 11 . . . 4 (𝑧𝐴 → ({𝑧} × (𝐹𝑧)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))})
87iuneq2i 4469 . . 3 𝑧𝐴 ({𝑧} × (𝐹𝑧)) = 𝑧𝐴 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))}
9 iunopab 4925 . . 3 𝑧𝐴 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))}
10 velsn 4140 . . . . . . . 8 (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
11 equcom 1931 . . . . . . . 8 (𝑥 = 𝑧𝑧 = 𝑥)
1210, 11bitri 262 . . . . . . 7 (𝑥 ∈ {𝑧} ↔ 𝑧 = 𝑥)
1312anbi1i 726 . . . . . 6 ((𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧)) ↔ (𝑧 = 𝑥𝑦 ∈ (𝐹𝑧)))
1413rexbii 3022 . . . . 5 (∃𝑧𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧)) ↔ ∃𝑧𝐴 (𝑧 = 𝑥𝑦 ∈ (𝐹𝑧)))
15 fveq2 6087 . . . . . . 7 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
1615eleq2d 2672 . . . . . 6 (𝑧 = 𝑥 → (𝑦 ∈ (𝐹𝑧) ↔ 𝑦 ∈ (𝐹𝑥)))
1716ceqsrexbv 3306 . . . . 5 (∃𝑧𝐴 (𝑧 = 𝑥𝑦 ∈ (𝐹𝑧)) ↔ (𝑥𝐴𝑦 ∈ (𝐹𝑥)))
1814, 17bitri 262 . . . 4 (∃𝑧𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧)) ↔ (𝑥𝐴𝑦 ∈ (𝐹𝑥)))
1918opabbii 4643 . . 3 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
208, 9, 193eqtri 2635 . 2 𝑧𝐴 ({𝑧} × (𝐹𝑧)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
211, 5, 203eqtri 2635 1 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1474  wcel 1976  wrex 2896  {csn 4124   ciun 4449  {copab 4636   × cxp 5025  cfv 5789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-xp 5033  df-iota 5753  df-fv 5797
This theorem is referenced by:  marypha2lem3  8203  marypha2lem4  8204  eulerpartlemgu  29559
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