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 (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 5824	. . 3 ⊢ (Rel 𝐹 → Rel (𝐹 ∖ 𝐴))
2		brdif 5195	. . . . . . 7 ⊢ (𝑥(𝐹 ∖ 𝐴)𝑦 ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐴𝑦))
3		brdif 5195	. . . . . . 7 ⊢ (𝑥(𝐹 ∖ 𝐴)𝑧 ↔ (𝑥𝐹𝑧 ∧ ¬ 𝑥𝐴𝑧))
4		pm2.27 42	. . . . . . . 8 ⊢ ((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → (((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) → 𝑦 = 𝑧))
5	4	ad2ant2r 747	. . . . . . 7 ⊢ (((𝑥𝐹𝑦 ∧ ¬ 𝑥𝐴𝑦) ∧ (𝑥𝐹𝑧 ∧ ¬ 𝑥𝐴𝑧)) → (((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) → 𝑦 = 𝑧))
6	2, 3, 5	syl2anb 598	. . . . . 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 1810	. . . 4 ⊢ (∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) → ∀𝑧((𝑥(𝐹 ∖ 𝐴)𝑦 ∧ 𝑥(𝐹 ∖ 𝐴)𝑧) → 𝑦 = 𝑧))
9	8	2alimi 1811	. . 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 613	. 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 6570	. 2 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
12		dffun2 6570	. 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 292	1 ⊢ (Fun 𝐹 → Fun (𝐹 ∖ 𝐴))

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