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Theorem fundif 6614
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 5824 . . 3 (Rel 𝐹 → Rel (𝐹𝐴))
2 brdif 5195 . . . . . . 7 (𝑥(𝐹𝐴)𝑦 ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐴𝑦))
3 brdif 5195 . . . . . . 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 1810 . . . 4 (∀𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧) → ∀𝑧((𝑥(𝐹𝐴)𝑦𝑥(𝐹𝐴)𝑧) → 𝑦 = 𝑧))
982alimi 1811 . . 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 6570 . 2 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥𝑦𝑧((𝑥𝐹𝑦𝑥𝐹𝑧) → 𝑦 = 𝑧)))
12 dffun2 6570 . 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 1537  cdif 3947   class class class wbr 5142  Rel wrel 5689  Fun wfun 6554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-fun 6562
This theorem is referenced by:  fundmge2nop  14543  fun2dmnop  14545
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