Theorem dffun3f 49720

Description: Alternate definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Emmett Weisz, 14-Mar-2021.)

Hypotheses

Ref	Expression
dffun3f.1	⊢ Ⅎ𝑥𝐴
dffun3f.2	⊢ Ⅎ𝑦𝐴
dffun3f.3	⊢ Ⅎ𝑧𝐴

Assertion

Ref	Expression
dffun3f	⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)))

Distinct variable group:   𝑥,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)

Proof of Theorem dffun3f

Step	Hyp	Ref	Expression
1		dffun3f.1	. . 3 ⊢ Ⅎ𝑥𝐴
2		dffun3f.2	. . 3 ⊢ Ⅎ𝑦𝐴
3	1, 2	dffun6f 6496	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
4		nfcv 2894	. . . . . 6 ⊢ Ⅎ𝑧𝑥
5		dffun3f.3	. . . . . 6 ⊢ Ⅎ𝑧𝐴
6		nfcv 2894	. . . . . 6 ⊢ Ⅎ𝑧𝑦
7	4, 5, 6	nfbr 5138	. . . . 5 ⊢ Ⅎ𝑧 𝑥𝐴𝑦
8	7	mof 2558	. . . 4 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ ∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))
9	8	albii 1820	. . 3 ⊢ (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))
10	9	anbi2i 623	. 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)))
11	3, 10	bitri 275	1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1539  ∃wex 1780  ∃*wmo 2533  Ⅎwnfc 2879   class class class wbr 5091  Rel wrel 5621  Fun wfun 6475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-fun 6483

This theorem is referenced by:  setrec2lem2  49732