Theorem dffun3f 9956

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 6546	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
4		nfcv 2923	. . . . . 6 ⊢ Ⅎ𝑧𝑥
5		dffun3f.3	. . . . . 6 ⊢ Ⅎ𝑧𝐴
6		nfcv 2923	. . . . . 6 ⊢ Ⅎ𝑧𝑦
7	4, 5, 6	nfbr 5152	. . . . 5 ⊢ Ⅎ𝑧 𝑥𝐴𝑦
8	7	mof 2589	. . . 4 ⊢ (∃*𝑦 𝑥𝐴𝑦 ↔ ∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))
9	8	albii 1852	. . 3 ⊢ (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))
10	9	anbi2i 635	. 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃wex 1812  ∃*wmo 2563  Ⅎwnfc 2908   class class class wbr 5103  Rel wrel 5656  Fun wfun 6525

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-fun 6533

This theorem is used by:  setrec2lem2  9957