Theorem dffun3f 50443

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 6553	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
4		nfcv 2925	. . . . . 6 ⊢ Ⅎ𝑧𝑥
5		dffun3f.3	. . . . . 6 ⊢ Ⅎ𝑧𝐴
6		nfcv 2925	. . . . . 6 ⊢ Ⅎ𝑧𝑦
7	4, 5, 6	nfbr 5159	. . . . 5 ⊢ Ⅎ𝑧 𝑥𝐴𝑦
8	7	mof 2591	. . . 4 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ ∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))
9	8	albii 1849	. . 3 ⊢ (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))
10	9	anbi2i 634	. 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)))
11	3, 10	bitri 278	1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568  ∃wex 1809  ∃*wmo 2565  Ⅎwnfc 2910   class class class wbr 5110  Rel wrel 5668  Fun wfun 6532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-fun 6540

This theorem is referenced by:  setrec2lem2  50455