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Theorem fundif 6391
 Description: A function with removed elements is still a function. (Contributed by AV, 7-Jun-2021.)
Assertion
Ref Expression
fundif (Fun 𝐹 → Fun (𝐹𝐴))

Proof of Theorem fundif
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reldif 5675 . . 3 (Rel 𝐹 → Rel (𝐹𝐴))
2 brdif 5105 . . . . . . 7 (𝑥(𝐹𝐴)𝑦 ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐴𝑦))
3 brdif 5105 . . . . . . 7 (𝑥(𝐹𝐴)𝑧 ↔ (𝑥𝐹𝑧 ∧ ¬ 𝑥𝐴𝑧))
4 pm2.27 42 . . . . . . . 8 ((𝑥𝐹𝑦𝑥𝐹𝑧) → (((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → 𝑦 = 𝑧))
54ad2ant2r 746 . . . . . . 7 (((𝑥𝐹𝑦 ∧ ¬ 𝑥𝐴𝑦) ∧ (𝑥𝐹𝑧 ∧ ¬ 𝑥𝐴𝑧)) → (((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → 𝑦 = 𝑧))
62, 3, 5syl2anb 600 . . . . . 6 ((𝑥(𝐹𝐴)𝑦𝑥(𝐹𝐴)𝑧) → (((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → 𝑦 = 𝑧))
76com12 32 . . . . 5 (((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → ((𝑥(𝐹𝐴)𝑦𝑥(𝐹𝐴)𝑧) → 𝑦 = 𝑧))
87alimi 1813 . . . 4 (∀𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → ∀𝑧((𝑥(𝐹𝐴)𝑦𝑥(𝐹𝐴)𝑧) → 𝑦 = 𝑧))
982alimi 1814 . . 3 (∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → ∀𝑥𝑦𝑧((𝑥(𝐹𝐴)𝑦𝑥(𝐹𝐴)𝑧) → 𝑦 = 𝑧))
101, 9anim12i 615 . 2 ((Rel 𝐹 ∧ ∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧)) → (Rel (𝐹𝐴) ∧ ∀𝑥𝑦𝑧((𝑥(𝐹𝐴)𝑦𝑥(𝐹𝐴)𝑧) → 𝑦 = 𝑧)))
11 dffun2 6353 . 2 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧)))
12 dffun2 6353 . 2 (Fun (𝐹𝐴) ↔ (Rel (𝐹𝐴) ∧ ∀𝑥𝑦𝑧((𝑥(𝐹𝐴)𝑦𝑥(𝐹𝐴)𝑧) → 𝑦 = 𝑧)))
1310, 11, 123imtr4i 295 1 (Fun 𝐹 → Fun (𝐹𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399  ∀wal 1536   ∖ cdif 3916   class class class wbr 5052  Rel wrel 5547  Fun wfun 6337 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317 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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-id 5447  df-rel 5549  df-cnv 5550  df-co 5551  df-fun 6345 This theorem is referenced by:  fundmge2nop  13856  fun2dmnop  13858
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