Theorem dffun6f 6363

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 6360	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑤∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢)))
2		nfcv 2977	. . . . . . 7 ⊢ Ⅎ𝑦𝑤
3		dffun6f.2	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
4		nfcv 2977	. . . . . . 7 ⊢ Ⅎ𝑦𝑣
5	2, 3, 4	nfbr 5105	. . . . . 6 ⊢ Ⅎ𝑦 𝑤𝐴𝑣
6		nfv 1911	. . . . . 6 ⊢ Ⅎ𝑣 𝑤𝐴𝑦
7		breq2 5062	. . . . . 6 ⊢ (𝑣 = 𝑦 → (𝑤𝐴𝑣 ↔ 𝑤𝐴𝑦))
8	5, 6, 7	cbvmow 2684	. . . . 5 ⊢ (∃*𝑣 𝑤𝐴𝑣 ↔ ∃*𝑦 𝑤𝐴𝑦)
9	8	albii 1816	. . . 4 ⊢ (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃*𝑦 𝑤𝐴𝑦)
10		df-mo 2618	. . . . 5 ⊢ (∃*𝑣 𝑤𝐴𝑣 ↔ ∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢))
11	10	albii 1816	. . . 4 ⊢ (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢))
12		nfcv 2977	. . . . . . 7 ⊢ Ⅎ𝑥𝑤
13		dffun6f.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
14		nfcv 2977	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
15	12, 13, 14	nfbr 5105	. . . . . 6 ⊢ Ⅎ𝑥 𝑤𝐴𝑦
16	15	nfmov 2640	. . . . 5 ⊢ Ⅎ𝑥∃*𝑦 𝑤𝐴𝑦
17		nfv 1911	. . . . 5 ⊢ Ⅎ𝑤∃*𝑦 𝑥𝐴𝑦
18		breq1 5061	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤𝐴𝑦 ↔ 𝑥𝐴𝑦))
19	18	mobidv 2629	. . . . 5 ⊢ (𝑤 = 𝑥 → (∃*𝑦 𝑤𝐴𝑦 ↔ ∃*𝑦 𝑥𝐴𝑦))
20	16, 17, 19	cbvalv1 2357	. . . 4 ⊢ (∀𝑤∃*𝑦 𝑤𝐴𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
21	9, 11, 20	3bitr3ri 304	. . 3 ⊢ (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑤∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢))
22	21	anbi2i 624	. 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑤∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢)))
23	1, 22	bitr4i 280	1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531  ∃wex 1776  ∃*wmo 2616  Ⅎwnfc 2961   class class class wbr 5058  Rel wrel 5554  Fun wfun 6343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-id 5454  df-cnv 5557  df-co 5558  df-fun 6351

This theorem is referenced by:  dffun6  6364  funopab  6384  funcnvmpt  30406  dffun3f  44779