Theorem dffun6f 6552

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 6549	. 2 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑤∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢)))
2		nfcv 2924	. . . . . . 7 ⊢ Ⅎ𝑦𝑤
3		dffun6f.2	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
4		nfcv 2924	. . . . . . 7 ⊢ Ⅎ𝑦𝑣
5	2, 3, 4	nfbr 5156	. . . . . 6 ⊢ Ⅎ𝑦 𝑤𝐴𝑣
6		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑣 𝑤𝐴𝑦
7		breq2 5111	. . . . . 6 ⊢ (𝑣 = 𝑦 → (𝑤𝐴𝑣 ↔ 𝑤𝐴𝑦))
8	5, 6, 7	cbvmow 2630	. . . . 5 ⊢ (∃*𝑣 𝑤𝐴𝑣 ↔ ∃*𝑦 𝑤𝐴𝑦)
9	8	albii 1852	. . . 4 ⊢ (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃*𝑦 𝑤𝐴𝑦)
10		dfmo 2567	. . . . 5 ⊢ (∃*𝑣 𝑤𝐴𝑣 ↔ ∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢))
11	10	albii 1852	. . . 4 ⊢ (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢))
12		nfcv 2924	. . . . . . 7 ⊢ Ⅎ𝑥𝑤
13		dffun6f.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
14		nfcv 2924	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
15	12, 13, 14	nfbr 5156	. . . . . 6 ⊢ Ⅎ𝑥 𝑤𝐴𝑦
16	15	nfmov 2587	. . . . 5 ⊢ Ⅎ𝑥∃*𝑦 𝑤𝐴𝑦
17		nfv 1947	. . . . 5 ⊢ Ⅎ𝑤∃*𝑦 𝑥𝐴𝑦
18		breq1 5110	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝑤𝐴𝑦 ↔ 𝑥𝐴𝑦))
19	18	mobidv 2576	. . . . 5 ⊢ (𝑤 = 𝑥 → (∃*𝑦 𝑤𝐴𝑦 ↔ ∃*𝑦 𝑥𝐴𝑦))
20	16, 17, 19	cbvalv1 2372	. . . 4 ⊢ (∀𝑤∃*𝑦 𝑤𝐴𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
21	9, 11, 20	3bitr3ri 305	. . 3 ⊢ (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑤∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢))
22	21	anbi2i 635	. 2 ⊢ ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑤∃𝑢∀𝑣(𝑤𝐴𝑣 → 𝑣 = 𝑢)))
23	1, 22	bitr4i 281	1 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃wex 1812  ∃*wmo 2564  Ⅎwnfc 2909   class class class wbr 5107  Rel wrel 5664  Fun wfun 6531

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-fun 6539

This theorem is used by:  funopab  6572  funcnvmpt  6992  dffun3f  50616