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Theorem marypha2lem2 8888
Description: Lemma for marypha2 8891. 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 4549 . . . 4 (𝑥 = 𝑧 → {𝑥} = {𝑧})
3 fveq2 6652 . . . 4 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
42, 3xpeq12d 5563 . . 3 (𝑥 = 𝑧 → ({𝑥} × (𝐹𝑥)) = ({𝑧} × (𝐹𝑧)))
54cbviunv 4940 . 2 𝑥𝐴 ({𝑥} × (𝐹𝑥)) = 𝑧𝐴 ({𝑧} × (𝐹𝑧))
6 df-xp 5538 . . . . 5 ({𝑧} × (𝐹𝑧)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))}
76a1i 11 . . . 4 (𝑧𝐴 → ({𝑧} × (𝐹𝑧)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))})
87iuneq2i 4915 . . 3 𝑧𝐴 ({𝑧} × (𝐹𝑧)) = 𝑧𝐴 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))}
9 iunopab 5423 . . 3 𝑧𝐴 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))}
10 velsn 4555 . . . . . . . 8 (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
11 equcom 2025 . . . . . . . 8 (𝑥 = 𝑧𝑧 = 𝑥)
1210, 11bitri 278 . . . . . . 7 (𝑥 ∈ {𝑧} ↔ 𝑧 = 𝑥)
1312anbi1i 626 . . . . . 6 ((𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧)) ↔ (𝑧 = 𝑥𝑦 ∈ (𝐹𝑧)))
1413rexbii 3235 . . . . 5 (∃𝑧𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧)) ↔ ∃𝑧𝐴 (𝑧 = 𝑥𝑦 ∈ (𝐹𝑧)))
15 fveq2 6652 . . . . . . 7 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
1615eleq2d 2899 . . . . . 6 (𝑧 = 𝑥 → (𝑦 ∈ (𝐹𝑧) ↔ 𝑦 ∈ (𝐹𝑥)))
1716ceqsrexbv 3625 . . . . 5 (∃𝑧𝐴 (𝑧 = 𝑥𝑦 ∈ (𝐹𝑧)) ↔ (𝑥𝐴𝑦 ∈ (𝐹𝑥)))
1814, 17bitri 278 . . . 4 (∃𝑧𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧)) ↔ (𝑥𝐴𝑦 ∈ (𝐹𝑥)))
1918opabbii 5109 . . 3 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 (𝑥 ∈ {𝑧} ∧ 𝑦 ∈ (𝐹𝑧))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
208, 9, 193eqtri 2849 . 2 𝑧𝐴 ({𝑧} × (𝐹𝑧)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
211, 5, 203eqtri 2849 1 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))}
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wcel 2114  wrex 3131  {csn 4539   ciun 4894  {copab 5104   × cxp 5530  cfv 6334
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-xp 5538  df-iota 6293  df-fv 6342
This theorem is referenced by:  marypha2lem3  8889  marypha2lem4  8890  eulerpartlemgu  31709
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