Theorem dffun6f 6591

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 6587	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑤∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢)))
2		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑦𝑤
3		dffun6f.2	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
4		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑦𝑣
5	2, 3, 4	nfbr 5213	. . . . . 6 ⊢ Ⅎ𝑦 𝑤𝐴𝑣
6		nfv 1913	. . . . . 6 ⊢ Ⅎ𝑣 𝑤𝐴𝑦
7		breq2 5170	. . . . . 6 ⊢ (𝑣 = 𝑦 → (𝑤𝐴𝑣 ↔ 𝑤𝐴𝑦))
8	5, 6, 7	cbvmow 2606	. . . . 5 ⊢ (∃*𝑣 𝑤𝐴𝑣 ↔ ∃*𝑦 𝑤𝐴𝑦)
9	8	albii 1817	. . . 4 ⊢ (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃*𝑦 𝑤𝐴𝑦)
10		df-mo 2543	. . . . 5 ⊢ (∃*𝑣 𝑤𝐴𝑣 ↔ ∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢))
11	10	albii 1817	. . . 4 ⊢ (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢))
12		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑥𝑤
13		dffun6f.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
14		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
15	12, 13, 14	nfbr 5213	. . . . . 6 ⊢ Ⅎ𝑥 𝑤𝐴𝑦
16	15	nfmov 2563	. . . . 5 ⊢ Ⅎ𝑥∃*𝑦 𝑤𝐴𝑦
17		nfv 1913	. . . . 5 ⊢ Ⅎ𝑤∃*𝑦 𝑥𝐴𝑦
18		breq1 5169	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤𝐴𝑦 ↔ 𝑥𝐴𝑦))
19	18	mobidv 2552	. . . . 5 ⊢ (𝑤 = 𝑥 → (∃*𝑦 𝑤𝐴𝑦 ↔ ∃*𝑦 𝑥𝐴𝑦))
20	16, 17, 19	cbvalv1 2347	. . . 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 622	. 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 1535  ∃wex 1777  ∃*wmo 2541  Ⅎwnfc 2893   class class class wbr 5166  Rel wrel 5705  Fun wfun 6567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-fun 6575

This theorem is referenced by:  dffun6OLD  6592  funopab  6613  funcnvmpt  32685  dffun3f  48774