Theorem dffun6f 5211

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 5208	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑤∀𝑣∀𝑢((𝑤𝐴𝑣 ∧ 𝑤𝐴𝑢) → 𝑣 = 𝑢)))
2		nfcv 2312	. . . . . . 7 ⊢ Ⅎ𝑦𝑤
3		dffun6f.2	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
4		nfcv 2312	. . . . . . 7 ⊢ Ⅎ𝑦𝑣
5	2, 3, 4	nfbr 4035	. . . . . 6 ⊢ Ⅎ𝑦 𝑤𝐴𝑣
6		nfv 1521	. . . . . 6 ⊢ Ⅎ𝑣 𝑤𝐴𝑦
7		breq2 3993	. . . . . 6 ⊢ (𝑣 = 𝑦 → (𝑤𝐴𝑣 ↔ 𝑤𝐴𝑦))
8	5, 6, 7	cbvmo 2059	. . . . 5 ⊢ (∃*𝑣 𝑤𝐴𝑣 ↔ ∃*𝑦 𝑤𝐴𝑦)
9	8	albii 1463	. . . 4 ⊢ (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃*𝑦 𝑤𝐴𝑦)
10		breq2 3993	. . . . . 6 ⊢ (𝑣 = 𝑢 → (𝑤𝐴𝑣 ↔ 𝑤𝐴𝑢))
11	10	mo4 2080	. . . . 5 ⊢ (∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑣∀𝑢((𝑤𝐴𝑣 ∧ 𝑤𝐴𝑢) → 𝑣 = 𝑢))
12	11	albii 1463	. . . 4 ⊢ (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∀𝑣∀𝑢((𝑤𝐴𝑣 ∧ 𝑤𝐴𝑢) → 𝑣 = 𝑢))
13		nfcv 2312	. . . . . . 7 ⊢ Ⅎ𝑥𝑤
14		dffun6f.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
15		nfcv 2312	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
16	13, 14, 15	nfbr 4035	. . . . . 6 ⊢ Ⅎ𝑥 𝑤𝐴𝑦
17	16	nfmo 2039	. . . . 5 ⊢ Ⅎ𝑥∃*𝑦 𝑤𝐴𝑦
18		nfv 1521	. . . . 5 ⊢ Ⅎ𝑤∃*𝑦 𝑥𝐴𝑦
19		breq1 3992	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤𝐴𝑦 ↔ 𝑥𝐴𝑦))
20	19	mobidv 2055	. . . . 5 ⊢ (𝑤 = 𝑥 → (∃*𝑦 𝑤𝐴𝑦 ↔ ∃*𝑦 𝑥𝐴𝑦))
21	17, 18, 20	cbval 1747	. . . 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 454	. 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 1346  ∃*wmo 2020  Ⅎwnfc 2299   class class class wbr 3989  Rel wrel 4616  Fun wfun 5192

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-id 4278  df-cnv 4619  df-co 4620  df-fun 5200

This theorem is referenced by:  dffun6  5212  dffun4f  5214  funopab  5233