Theorem fundif 6406

Description: A function with removed elements is still a function. (Contributed by AV, 7-Jun-2021.)

Assertion

Ref	Expression
fundif	⊢ (Fun 𝐹 → Fun (𝐹 ∖ 𝐴))

Proof of Theorem fundif

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reldif 5691	. . 3 ⊢ (Rel 𝐹 → Rel (𝐹 ∖ 𝐴))
2		brdif 5122	. . . . . . 7 ⊢ (𝑥(𝐹 ∖ 𝐴)𝑦 ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐴𝑦))
3		brdif 5122	. . . . . . 7 ⊢ (𝑥(𝐹 ∖ 𝐴)𝑧 ↔ (𝑥𝐹𝑧 ∧ ¬ 𝑥𝐴𝑧))
4		pm2.27 42	. . . . . . . 8 ⊢ ((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → (((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) → 𝑦 = 𝑧))
5	4	ad2ant2r 745	. . . . . . 7 ⊢ (((𝑥𝐹𝑦 ∧ ¬ 𝑥𝐴𝑦) ∧ (𝑥𝐹𝑧 ∧ ¬ 𝑥𝐴𝑧)) → (((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) → 𝑦 = 𝑧))
6	2, 3, 5	syl2anb 599	. . . . . 6 ⊢ ((𝑥(𝐹 ∖ 𝐴)𝑦 ∧ 𝑥(𝐹 ∖ 𝐴)𝑧) → (((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) → 𝑦 = 𝑧))
7	6	com12 32	. . . . 5 ⊢ (((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) → ((𝑥(𝐹 ∖ 𝐴)𝑦 ∧ 𝑥(𝐹 ∖ 𝐴)𝑧) → 𝑦 = 𝑧))
8	7	alimi 1811	. . . 4 ⊢ (∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) → ∀𝑧((𝑥(𝐹 ∖ 𝐴)𝑦 ∧ 𝑥(𝐹 ∖ 𝐴)𝑧) → 𝑦 = 𝑧))
9	8	2alimi 1812	. . 3 ⊢ (∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) → ∀𝑥∀𝑦∀𝑧((𝑥(𝐹 ∖ 𝐴)𝑦 ∧ 𝑥(𝐹 ∖ 𝐴)𝑧) → 𝑦 = 𝑧))
10	1, 9	anim12i 614	. 2 ⊢ ((Rel 𝐹 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧)) → (Rel (𝐹 ∖ 𝐴) ∧ ∀𝑥∀𝑦∀𝑧((𝑥(𝐹 ∖ 𝐴)𝑦 ∧ 𝑥(𝐹 ∖ 𝐴)𝑧) → 𝑦 = 𝑧)))
11		dffun2 6368	. 2 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
12		dffun2 6368	. 2 ⊢ (Fun (𝐹 ∖ 𝐴) ↔ (Rel (𝐹 ∖ 𝐴) ∧ ∀𝑥∀𝑦∀𝑧((𝑥(𝐹 ∖ 𝐴)𝑦 ∧ 𝑥(𝐹 ∖ 𝐴)𝑧) → 𝑦 = 𝑧)))
13	10, 11, 12	3imtr4i 294	1 ⊢ (Fun 𝐹 → Fun (𝐹 ∖ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398  ∀wal 1534   ∖ cdif 3936   class class class wbr 5069  Rel wrel 5563  Fun wfun 6352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-br 5070  df-opab 5132  df-id 5463  df-rel 5565  df-cnv 5566  df-co 5567  df-fun 6360

This theorem is referenced by:  fundmge2nop  13854  fun2dmnop  13856