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Theorem funressnvmoOLD 42102
 Description: Old proof of funressnvmo 42101. Obsolete as of 9-Oct-2022. (Contributed by AV, 2-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
funressnvmoOLD (Fun (𝐹 ↾ {𝑥}) → ∃*𝑦 𝑥𝐹𝑦)
Distinct variable group:   𝑥,𝑦,𝐹

Proof of Theorem funressnvmoOLD
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dffun6 6150 . 2 (Fun (𝐹 ↾ {𝑥}) ↔ (Rel (𝐹 ↾ {𝑥}) ∧ ∀𝑧∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦))
2 breq1 4889 . . . . . . 7 (𝑥 = 𝑧 → (𝑥(𝐹 ↾ {𝑥})𝑦𝑧(𝐹 ↾ {𝑥})𝑦))
32equcoms 2066 . . . . . 6 (𝑧 = 𝑥 → (𝑥(𝐹 ↾ {𝑥})𝑦𝑧(𝐹 ↾ {𝑥})𝑦))
43biimpd 221 . . . . 5 (𝑧 = 𝑥 → (𝑥(𝐹 ↾ {𝑥})𝑦𝑧(𝐹 ↾ {𝑥})𝑦))
54moimdv 2558 . . . 4 (𝑧 = 𝑥 → (∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝑥})𝑦))
65spimvw 2045 . . 3 (∀𝑧∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥(𝐹 ↾ {𝑥})𝑦)
7 id 22 . . . . 5 (𝑥𝐹𝑦𝑥𝐹𝑦)
8 vsnid 4430 . . . . 5 𝑥 ∈ {𝑥}
9 vex 3400 . . . . . 6 𝑦 ∈ V
109brresOLD2 5655 . . . . 5 (𝑥(𝐹 ↾ {𝑥})𝑦 ↔ (𝑥𝐹𝑦𝑥 ∈ {𝑥}))
117, 8, 10sylanblrc 584 . . . 4 (𝑥𝐹𝑦𝑥(𝐹 ↾ {𝑥})𝑦)
1211moimi 2556 . . 3 (∃*𝑦 𝑥(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥𝐹𝑦)
136, 12syl 17 . 2 (∀𝑧∃*𝑦 𝑧(𝐹 ↾ {𝑥})𝑦 → ∃*𝑦 𝑥𝐹𝑦)
141, 13simplbiim 500 1 (Fun (𝐹 ↾ {𝑥}) → ∃*𝑦 𝑥𝐹𝑦)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1599   ∈ wcel 2106  ∃*wmo 2548  {csn 4397   class class class wbr 4886   ↾ cres 5357  Rel wrel 5360  Fun wfun 6129 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-cnv 5363  df-co 5364  df-res 5367  df-fun 6137 This theorem is referenced by: (None)
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