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Theorem fundif 6617
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 5828 . . 3 (Rel 𝐹 → Rel (𝐹𝐴))
2 brdif 5201 . . . . . . 7 (𝑥(𝐹𝐴)𝑦 ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐴𝑦))
3 brdif 5201 . . . . . . 7 (𝑥(𝐹𝐴)𝑧 ↔ (𝑥𝐹𝑧 ∧ ¬ 𝑥𝐴𝑧))
4 pm2.27 42 . . . . . . . 8 ((𝑥𝐹𝑦𝑥𝐹𝑧) → (((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → 𝑦 = 𝑧))
54ad2ant2r 747 . . . . . . 7 (((𝑥𝐹𝑦 ∧ ¬ 𝑥𝐴𝑦) ∧ (𝑥𝐹𝑧 ∧ ¬ 𝑥𝐴𝑧)) → (((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → 𝑦 = 𝑧))
62, 3, 5syl2anb 598 . . . . . 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 1808 . . . 4 (∀𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → ∀𝑧((𝑥(𝐹𝐴)𝑦𝑥(𝐹𝐴)𝑧) → 𝑦 = 𝑧))
982alimi 1809 . . 3 (∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → ∀𝑥𝑦𝑧((𝑥(𝐹𝐴)𝑦𝑥(𝐹𝐴)𝑧) → 𝑦 = 𝑧))
101, 9anim12i 613 . 2 ((Rel 𝐹 ∧ ∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧)) → (Rel (𝐹𝐴) ∧ ∀𝑥𝑦𝑧((𝑥(𝐹𝐴)𝑦𝑥(𝐹𝐴)𝑧) → 𝑦 = 𝑧)))
11 dffun2 6573 . 2 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧)))
12 dffun2 6573 . 2 (Fun (𝐹𝐴) ↔ (Rel (𝐹𝐴) ∧ ∀𝑥𝑦𝑧((𝑥(𝐹𝐴)𝑦𝑥(𝐹𝐴)𝑧) → 𝑦 = 𝑧)))
1310, 11, 123imtr4i 292 1 (Fun 𝐹 → Fun (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1535  cdif 3960   class class class wbr 5148  Rel wrel 5694  Fun wfun 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-fun 6565
This theorem is referenced by:  fundmge2nop  14539  fun2dmnop  14541
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