Theorem fundif 6467

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 5714	. . 3 ⊢ (Rel 𝐹 → Rel (𝐹 ∖ 𝐴))
2		brdif 5123	. . . . . . 7 ⊢ (𝑥(𝐹 ∖ 𝐴)𝑦 ↔ (𝑥𝐹𝑦 ∧ ¬ 𝑥𝐴𝑦))
3		brdif 5123	. . . . . . 7 ⊢ (𝑥(𝐹 ∖ 𝐴)𝑧 ↔ (𝑥𝐹𝑧 ∧ ¬ 𝑥𝐴𝑧))
4		pm2.27 42	. . . . . . . 8 ⊢ ((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → (((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) → 𝑦 = 𝑧))
5	4	ad2ant2r 743	. . . . . . 7 ⊢ (((𝑥𝐹𝑦 ∧ ¬ 𝑥𝐴𝑦) ∧ (𝑥𝐹𝑧 ∧ ¬ 𝑥𝐴𝑧)) → (((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) → 𝑦 = 𝑧))
6	2, 3, 5	syl2anb 597	. . . . . 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 1815	. . . 4 ⊢ (∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧) → ∀𝑧((𝑥(𝐹 ∖ 𝐴)𝑦 ∧ 𝑥(𝐹 ∖ 𝐴)𝑧) → 𝑦 = 𝑧))
9	8	2alimi 1816	. . 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 612	. 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 6428	. 2 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑦 ∧ 𝑥𝐹𝑧) → 𝑦 = 𝑧)))
12		dffun2 6428	. 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 291	1 ⊢ (Fun 𝐹 → Fun (𝐹 ∖ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395  ∀wal 1537   ∖ cdif 3880   class class class wbr 5070  Rel wrel 5585  Fun wfun 6412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-id 5480  df-rel 5587  df-cnv 5588  df-co 5589  df-fun 6420

This theorem is referenced by:  fundmge2nop  14135  fun2dmnop  14137