Theorem dffun3f 43324

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 6137	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
4		nfcv 2969	. . . . . 6 ⊢ Ⅎ𝑧𝑥
5		dffun3f.3	. . . . . 6 ⊢ Ⅎ𝑧𝐴
6		nfcv 2969	. . . . . 6 ⊢ Ⅎ𝑧𝑦
7	4, 5, 6	nfbr 4920	. . . . 5 ⊢ Ⅎ𝑧 𝑥𝐴𝑦
8	7	mof 2631	. . . 4 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ ∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))
9	8	albii 1920	. . 3 ⊢ (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))
10	9	anbi2i 618	. 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)))
11	3, 10	bitri 267	1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1656  ∃wex 1880  ∃*wmo 2603  Ⅎwnfc 2956   class class class wbr 4873  Rel wrel 5347  Fun wfun 6117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-id 5250  df-cnv 5350  df-co 5351  df-fun 6125

This theorem is referenced by:  setrec2lem2  43336