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Theorem fundif 6096
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 5394 . . 3 (Rel 𝐹 → Rel (𝐹𝐴))
2 brdif 4857 . . . . . . 7 (𝑥(𝐹𝐴)𝑦 ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐴𝑦))
3 brdif 4857 . . . . . . 7 (𝑥(𝐹𝐴)𝑧 ↔ (𝑥𝐹𝑧 ∧ ¬ 𝑥𝐴𝑧))
4 pm2.27 42 . . . . . . . 8 ((𝑥𝐹𝑦𝑥𝐹𝑧) → (((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → 𝑦 = 𝑧))
54ad2ant2r 800 . . . . . . 7 (((𝑥𝐹𝑦 ∧ ¬ 𝑥𝐴𝑦) ∧ (𝑥𝐹𝑧 ∧ ¬ 𝑥𝐴𝑧)) → (((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → 𝑦 = 𝑧))
62, 3, 5syl2anb 497 . . . . . 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 1888 . . . 4 (∀𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → ∀𝑧((𝑥(𝐹𝐴)𝑦𝑥(𝐹𝐴)𝑧) → 𝑦 = 𝑧))
982alimi 1889 . . 3 (∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → ∀𝑥𝑦𝑧((𝑥(𝐹𝐴)𝑦𝑥(𝐹𝐴)𝑧) → 𝑦 = 𝑧))
101, 9anim12i 591 . 2 ((Rel 𝐹 ∧ ∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧)) → (Rel (𝐹𝐴) ∧ ∀𝑥𝑦𝑧((𝑥(𝐹𝐴)𝑦𝑥(𝐹𝐴)𝑧) → 𝑦 = 𝑧)))
11 dffun2 6059 . 2 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧)))
12 dffun2 6059 . 2 (Fun (𝐹𝐴) ↔ (Rel (𝐹𝐴) ∧ ∀𝑥𝑦𝑧((𝑥(𝐹𝐴)𝑦𝑥(𝐹𝐴)𝑧) → 𝑦 = 𝑧)))
1310, 11, 123imtr4i 281 1 (Fun 𝐹 → Fun (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wal 1630  cdif 3712   class class class wbr 4804  Rel wrel 5271  Fun wfun 6043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-id 5174  df-rel 5273  df-cnv 5274  df-co 5275  df-fun 6051
This theorem is referenced by:  fundmge2nop  13467  fun2dmnop  13469  basvtxvalOLD  26102  edgfiedgvalOLD  26103
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