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Theorem fundif 6149
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 5442 . . 3 (Rel 𝐹 → Rel (𝐹𝐴))
2 brdif 4896 . . . . . . 7 (𝑥(𝐹𝐴)𝑦 ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐴𝑦))
3 brdif 4896 . . . . . . 7 (𝑥(𝐹𝐴)𝑧 ↔ (𝑥𝐹𝑧 ∧ ¬ 𝑥𝐴𝑧))
4 pm2.27 42 . . . . . . . 8 ((𝑥𝐹𝑦𝑥𝐹𝑧) → (((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → 𝑦 = 𝑧))
54ad2ant2r 754 . . . . . . 7 (((𝑥𝐹𝑦 ∧ ¬ 𝑥𝐴𝑦) ∧ (𝑥𝐹𝑧 ∧ ¬ 𝑥𝐴𝑧)) → (((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → 𝑦 = 𝑧))
62, 3, 5syl2anb 592 . . . . . 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 1907 . . . 4 (∀𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → ∀𝑧((𝑥(𝐹𝐴)𝑦𝑥(𝐹𝐴)𝑧) → 𝑦 = 𝑧))
982alimi 1908 . . 3 (∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → ∀𝑥𝑦𝑧((𝑥(𝐹𝐴)𝑦𝑥(𝐹𝐴)𝑧) → 𝑦 = 𝑧))
101, 9anim12i 607 . 2 ((Rel 𝐹 ∧ ∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧)) → (Rel (𝐹𝐴) ∧ ∀𝑥𝑦𝑧((𝑥(𝐹𝐴)𝑦𝑥(𝐹𝐴)𝑧) → 𝑦 = 𝑧)))
11 dffun2 6111 . 2 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧)))
12 dffun2 6111 . 2 (Fun (𝐹𝐴) ↔ (Rel (𝐹𝐴) ∧ ∀𝑥𝑦𝑧((𝑥(𝐹𝐴)𝑦𝑥(𝐹𝐴)𝑧) → 𝑦 = 𝑧)))
1310, 11, 123imtr4i 284 1 (Fun 𝐹 → Fun (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385  wal 1651  cdif 3766   class class class wbr 4843  Rel wrel 5317  Fun wfun 6095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-id 5220  df-rel 5319  df-cnv 5320  df-co 5321  df-fun 6103
This theorem is referenced by:  fundmge2nop  13524  fun2dmnop  13526
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