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Theorem fununmo 5896
Description: If the union of classes is a function, there is at most one element in relation to an arbitrary element regarding one of these classes. (Contributed by AV, 18-Jul-2019.)
Assertion
Ref Expression
fununmo (Fun (𝐹𝐺) → ∃*𝑦 𝑥𝐹𝑦)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐹   𝑦,𝐺
Allowed substitution hints:   𝐹(𝑥)   𝐺(𝑥)

Proof of Theorem fununmo
StepHypRef Expression
1 funmo 5868 . 2 (Fun (𝐹𝐺) → ∃*𝑦 𝑥(𝐹𝐺)𝑦)
2 orc 400 . . . 4 (𝑥𝐹𝑦 → (𝑥𝐹𝑦𝑥𝐺𝑦))
3 brun 4668 . . . 4 (𝑥(𝐹𝐺)𝑦 ↔ (𝑥𝐹𝑦𝑥𝐺𝑦))
42, 3sylibr 224 . . 3 (𝑥𝐹𝑦𝑥(𝐹𝐺)𝑦)
54moimi 2519 . 2 (∃*𝑦 𝑥(𝐹𝐺)𝑦 → ∃*𝑦 𝑥𝐹𝑦)
61, 5syl 17 1 (Fun (𝐹𝐺) → ∃*𝑦 𝑥𝐹𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383  ∃*wmo 2470  cun 3557   class class class wbr 4618  Fun wfun 5846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-fun 5854
This theorem is referenced by:  fununfun  5897
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