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Theorem dffun6f 6151
 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 𝑥𝐴
dffun6f.2 𝑦𝐴
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.
StepHypRef Expression
1 dffun3 6148 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑤𝑢𝑣(𝑤𝐴𝑣𝑣 = 𝑢)))
2 nfcv 2934 . . . . . . 7 𝑦𝑤
3 dffun6f.2 . . . . . . 7 𝑦𝐴
4 nfcv 2934 . . . . . . 7 𝑦𝑣
52, 3, 4nfbr 4935 . . . . . 6 𝑦 𝑤𝐴𝑣
6 nfv 1957 . . . . . 6 𝑣 𝑤𝐴𝑦
7 breq2 4892 . . . . . 6 (𝑣 = 𝑦 → (𝑤𝐴𝑣𝑤𝐴𝑦))
85, 6, 7cbvmo 2637 . . . . 5 (∃*𝑣 𝑤𝐴𝑣 ↔ ∃*𝑦 𝑤𝐴𝑦)
98albii 1863 . . . 4 (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃*𝑦 𝑤𝐴𝑦)
10 df-mo 2551 . . . . 5 (∃*𝑣 𝑤𝐴𝑣 ↔ ∃𝑢𝑣(𝑤𝐴𝑣𝑣 = 𝑢))
1110albii 1863 . . . 4 (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤𝑢𝑣(𝑤𝐴𝑣𝑣 = 𝑢))
12 nfcv 2934 . . . . . . 7 𝑥𝑤
13 dffun6f.1 . . . . . . 7 𝑥𝐴
14 nfcv 2934 . . . . . . 7 𝑥𝑦
1512, 13, 14nfbr 4935 . . . . . 6 𝑥 𝑤𝐴𝑦
1615nfmo 2577 . . . . 5 𝑥∃*𝑦 𝑤𝐴𝑦
17 nfv 1957 . . . . 5 𝑤∃*𝑦 𝑥𝐴𝑦
18 breq1 4891 . . . . . 6 (𝑤 = 𝑥 → (𝑤𝐴𝑦𝑥𝐴𝑦))
1918mobidv 2564 . . . . 5 (𝑤 = 𝑥 → (∃*𝑦 𝑤𝐴𝑦 ↔ ∃*𝑦 𝑥𝐴𝑦))
2016, 17, 19cbvalv1 2312 . . . 4 (∀𝑤∃*𝑦 𝑤𝐴𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
219, 11, 203bitr3ri 294 . . 3 (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑤𝑢𝑣(𝑤𝐴𝑣𝑣 = 𝑢))
2221anbi2i 616 . 2 ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑤𝑢𝑣(𝑤𝐴𝑣𝑣 = 𝑢)))
231, 22bitr4i 270 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386  ∀wal 1599  ∃wex 1823  ∃*wmo 2549  Ⅎwnfc 2919   class class class wbr 4888  Rel wrel 5362  Fun wfun 6131 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4889  df-opab 4951  df-id 5263  df-cnv 5365  df-co 5366  df-fun 6139 This theorem is referenced by:  dffun6  6152  funopab  6172  funcnvmpt  30036  dffun3f  43544
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