Theorem dffun6f 5201

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 5198	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑤∀𝑣∀𝑢((𝑤𝐴𝑣 ∧ 𝑤𝐴𝑢) → 𝑣 = 𝑢)))
2		nfcv 2308	. . . . . . 7 ⊢ Ⅎ𝑦𝑤
3		dffun6f.2	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
4		nfcv 2308	. . . . . . 7 ⊢ Ⅎ𝑦𝑣
5	2, 3, 4	nfbr 4028	. . . . . 6 ⊢ Ⅎ𝑦 𝑤𝐴𝑣
6		nfv 1516	. . . . . 6 ⊢ Ⅎ𝑣 𝑤𝐴𝑦
7		breq2 3986	. . . . . 6 ⊢ (𝑣 = 𝑦 → (𝑤𝐴𝑣 ↔ 𝑤𝐴𝑦))
8	5, 6, 7	cbvmo 2054	. . . . 5 ⊢ (∃*𝑣 𝑤𝐴𝑣 ↔ ∃*𝑦 𝑤𝐴𝑦)
9	8	albii 1458	. . . 4 ⊢ (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃*𝑦 𝑤𝐴𝑦)
10		breq2 3986	. . . . . 6 ⊢ (𝑣 = 𝑢 → (𝑤𝐴𝑣 ↔ 𝑤𝐴𝑢))
11	10	mo4 2075	. . . . 5 ⊢ (∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑣∀𝑢((𝑤𝐴𝑣 ∧ 𝑤𝐴𝑢) → 𝑣 = 𝑢))
12	11	albii 1458	. . . 4 ⊢ (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∀𝑣∀𝑢((𝑤𝐴𝑣 ∧ 𝑤𝐴𝑢) → 𝑣 = 𝑢))
13		nfcv 2308	. . . . . . 7 ⊢ Ⅎ𝑥𝑤
14		dffun6f.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
15		nfcv 2308	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
16	13, 14, 15	nfbr 4028	. . . . . 6 ⊢ Ⅎ𝑥 𝑤𝐴𝑦
17	16	nfmo 2034	. . . . 5 ⊢ Ⅎ𝑥∃*𝑦 𝑤𝐴𝑦
18		nfv 1516	. . . . 5 ⊢ Ⅎ𝑤∃*𝑦 𝑥𝐴𝑦
19		breq1 3985	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤𝐴𝑦 ↔ 𝑥𝐴𝑦))
20	19	mobidv 2050	. . . . 5 ⊢ (𝑤 = 𝑥 → (∃*𝑦 𝑤𝐴𝑦 ↔ ∃*𝑦 𝑥𝐴𝑦))
21	17, 18, 20	cbval 1742	. . . 4 ⊢ (∀𝑤∃*𝑦 𝑤𝐴𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
22	9, 12, 21	3bitr3ri 210	. . 3 ⊢ (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑤∀𝑣∀𝑢((𝑤𝐴𝑣 ∧ 𝑤𝐴𝑢) → 𝑣 = 𝑢))
23	22	anbi2i 453	. 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑤∀𝑣∀𝑢((𝑤𝐴𝑣 ∧ 𝑤𝐴𝑢) → 𝑣 = 𝑢)))
24	1, 23	bitr4i 186	1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1341  ∃*wmo 2015  Ⅎwnfc 2295   class class class wbr 3982  Rel wrel 4609  Fun wfun 5182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-cnv 4612  df-co 4613  df-fun 5190

This theorem is referenced by:  dffun6  5202  dffun4f  5204  funopab  5223