Theorem dffun6f 5231

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		dffun2 5228	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑤∀𝑣∀𝑢((𝑤𝐴𝑣 ∧ 𝑤𝐴𝑢) → 𝑣 = 𝑢)))
2		nfcv 2319	. . . . . . 7 ⊢ Ⅎ𝑦𝑤
3		dffun6f.2	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
4		nfcv 2319	. . . . . . 7 ⊢ Ⅎ𝑦𝑣
5	2, 3, 4	nfbr 4051	. . . . . 6 ⊢ Ⅎ𝑦 𝑤𝐴𝑣
6		nfv 1528	. . . . . 6 ⊢ Ⅎ𝑣 𝑤𝐴𝑦
7		breq2 4009	. . . . . 6 ⊢ (𝑣 = 𝑦 → (𝑤𝐴𝑣 ↔ 𝑤𝐴𝑦))
8	5, 6, 7	cbvmo 2066	. . . . 5 ⊢ (∃*𝑣 𝑤𝐴𝑣 ↔ ∃*𝑦 𝑤𝐴𝑦)
9	8	albii 1470	. . . 4 ⊢ (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃*𝑦 𝑤𝐴𝑦)
10		breq2 4009	. . . . . 6 ⊢ (𝑣 = 𝑢 → (𝑤𝐴𝑣 ↔ 𝑤𝐴𝑢))
11	10	mo4 2087	. . . . 5 ⊢ (∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑣∀𝑢((𝑤𝐴𝑣 ∧ 𝑤𝐴𝑢) → 𝑣 = 𝑢))
12	11	albii 1470	. . . 4 ⊢ (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∀𝑣∀𝑢((𝑤𝐴𝑣 ∧ 𝑤𝐴𝑢) → 𝑣 = 𝑢))
13		nfcv 2319	. . . . . . 7 ⊢ Ⅎ𝑥𝑤
14		dffun6f.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
15		nfcv 2319	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
16	13, 14, 15	nfbr 4051	. . . . . 6 ⊢ Ⅎ𝑥 𝑤𝐴𝑦
17	16	nfmo 2046	. . . . 5 ⊢ Ⅎ𝑥∃*𝑦 𝑤𝐴𝑦
18		nfv 1528	. . . . 5 ⊢ Ⅎ𝑤∃*𝑦 𝑥𝐴𝑦
19		breq1 4008	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤𝐴𝑦 ↔ 𝑥𝐴𝑦))
20	19	mobidv 2062	. . . . 5 ⊢ (𝑤 = 𝑥 → (∃*𝑦 𝑤𝐴𝑦 ↔ ∃*𝑦 𝑥𝐴𝑦))
21	17, 18, 20	cbval 1754	. . . 4 ⊢ (∀𝑤∃*𝑦 𝑤𝐴𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
22	9, 12, 21	3bitr3ri 211	. . 3 ⊢ (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑤∀𝑣∀𝑢((𝑤𝐴𝑣 ∧ 𝑤𝐴𝑢) → 𝑣 = 𝑢))
23	22	anbi2i 457	. 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑤∀𝑣∀𝑢((𝑤𝐴𝑣 ∧ 𝑤𝐴𝑢) → 𝑣 = 𝑢)))
24	1, 23	bitr4i 187	1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1351  ∃*wmo 2027  Ⅎwnfc 2306   class class class wbr 4005  Rel wrel 4633  Fun wfun 5212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-id 4295  df-cnv 4636  df-co 4637  df-fun 5220

This theorem is referenced by:  dffun6  5232  dffun4f  5234  funopab  5253