Theorem dffun6f 6579

Description: Definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
dffun6f.1	⊢ Ⅎ𝑥𝐴
dffun6f.2	⊢ Ⅎ𝑦𝐴

Assertion

Ref	Expression
dffun6f	⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem dffun6f

Dummy variables 𝑤 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffun3 6575	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑤∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢)))
2		nfcv 2905	. . . . . . 7 ⊢ Ⅎ𝑦𝑤
3		dffun6f.2	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
4		nfcv 2905	. . . . . . 7 ⊢ Ⅎ𝑦𝑣
5	2, 3, 4	nfbr 5190	. . . . . 6 ⊢ Ⅎ𝑦 𝑤𝐴𝑣
6		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑣 𝑤𝐴𝑦
7		breq2 5147	. . . . . 6 ⊢ (𝑣 = 𝑦 → (𝑤𝐴𝑣 ↔ 𝑤𝐴𝑦))
8	5, 6, 7	cbvmow 2603	. . . . 5 ⊢ (∃*𝑣 𝑤𝐴𝑣 ↔ ∃*𝑦 𝑤𝐴𝑦)
9	8	albii 1819	. . . 4 ⊢ (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃*𝑦 𝑤𝐴𝑦)
10		df-mo 2540	. . . . 5 ⊢ (∃*𝑣 𝑤𝐴𝑣 ↔ ∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢))
11	10	albii 1819	. . . 4 ⊢ (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢))
12		nfcv 2905	. . . . . . 7 ⊢ Ⅎ𝑥𝑤
13		dffun6f.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
14		nfcv 2905	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
15	12, 13, 14	nfbr 5190	. . . . . 6 ⊢ Ⅎ𝑥 𝑤𝐴𝑦
16	15	nfmov 2560	. . . . 5 ⊢ Ⅎ𝑥∃*𝑦 𝑤𝐴𝑦
17		nfv 1914	. . . . 5 ⊢ Ⅎ𝑤∃*𝑦 𝑥𝐴𝑦
18		breq1 5146	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤𝐴𝑦 ↔ 𝑥𝐴𝑦))
19	18	mobidv 2549	. . . . 5 ⊢ (𝑤 = 𝑥 → (∃*𝑦 𝑤𝐴𝑦 ↔ ∃*𝑦 𝑥𝐴𝑦))
20	16, 17, 19	cbvalv1 2343	. . . 4 ⊢ (∀𝑤∃*𝑦 𝑤𝐴𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
21	9, 11, 20	3bitr3ri 302	. . 3 ⊢ (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑤∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢))
22	21	anbi2i 623	. 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑤∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢)))
23	1, 22	bitr4i 278	1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538  ∃wex 1779  ∃*wmo 2538  Ⅎwnfc 2890   class class class wbr 5143  Rel wrel 5690  Fun wfun 6555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-fun 6563

This theorem is referenced by:  dffun6OLD  6580  funopab  6601  funcnvmpt  32677  dffun3f  49201