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Theorem funmoOLD 6564
Description: Obsolete version of funmo 6563 as of 19-Dec-2024. (Contributed by NM, 24-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
funmoOLD (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐹

Proof of Theorem funmoOLD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dffun6 6556 . . . . . 6 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
21simplbi 498 . . . . 5 (Fun 𝐹 → Rel 𝐹)
3 brrelex1 5729 . . . . . 6 ((Rel 𝐹𝐴𝐹𝑦) → 𝐴 ∈ V)
43ex 413 . . . . 5 (Rel 𝐹 → (𝐴𝐹𝑦𝐴 ∈ V))
52, 4syl 17 . . . 4 (Fun 𝐹 → (𝐴𝐹𝑦𝐴 ∈ V))
65ancrd 552 . . 3 (Fun 𝐹 → (𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦)))
76alrimiv 1930 . 2 (Fun 𝐹 → ∀𝑦(𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦)))
8 breq1 5151 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
98mobidv 2543 . . . . . 6 (𝑥 = 𝐴 → (∃*𝑦 𝑥𝐹𝑦 ↔ ∃*𝑦 𝐴𝐹𝑦))
109imbi2d 340 . . . . 5 (𝑥 = 𝐴 → ((Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦) ↔ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)))
111simprbi 497 . . . . . 6 (Fun 𝐹 → ∀𝑥∃*𝑦 𝑥𝐹𝑦)
121119.21bi 2182 . . . . 5 (Fun 𝐹 → ∃*𝑦 𝑥𝐹𝑦)
1310, 12vtoclg 3556 . . . 4 (𝐴 ∈ V → (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦))
1413com12 32 . . 3 (Fun 𝐹 → (𝐴 ∈ V → ∃*𝑦 𝐴𝐹𝑦))
15 moanimv 2615 . . 3 (∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦) ↔ (𝐴 ∈ V → ∃*𝑦 𝐴𝐹𝑦))
1614, 15sylibr 233 . 2 (Fun 𝐹 → ∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦))
17 moim 2538 . 2 (∀𝑦(𝐴𝐹𝑦 → (𝐴 ∈ V ∧ 𝐴𝐹𝑦)) → (∃*𝑦(𝐴 ∈ V ∧ 𝐴𝐹𝑦) → ∃*𝑦 𝐴𝐹𝑦))
187, 16, 17sylc 65 1 (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1539   = wceq 1541  wcel 2106  ∃*wmo 2532  Vcvv 3474   class class class wbr 5148  Rel wrel 5681  Fun wfun 6537
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-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
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-sb 2068  df-mo 2534  df-clab 2710  df-cleq 2724  df-clel 2810  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-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-fun 6545
This theorem is referenced by: (None)
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