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Theorem dffun6f 6348
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 6345 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑤𝑢𝑣(𝑤𝐴𝑣𝑣 = 𝑢)))
2 nfcv 2979 . . . . . . 7 𝑦𝑤
3 dffun6f.2 . . . . . . 7 𝑦𝐴
4 nfcv 2979 . . . . . . 7 𝑦𝑣
52, 3, 4nfbr 5089 . . . . . 6 𝑦 𝑤𝐴𝑣
6 nfv 1915 . . . . . 6 𝑣 𝑤𝐴𝑦
7 breq2 5046 . . . . . 6 (𝑣 = 𝑦 → (𝑤𝐴𝑣𝑤𝐴𝑦))
85, 6, 7cbvmow 2687 . . . . 5 (∃*𝑣 𝑤𝐴𝑣 ↔ ∃*𝑦 𝑤𝐴𝑦)
98albii 1821 . . . 4 (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃*𝑦 𝑤𝐴𝑦)
10 df-mo 2622 . . . . 5 (∃*𝑣 𝑤𝐴𝑣 ↔ ∃𝑢𝑣(𝑤𝐴𝑣𝑣 = 𝑢))
1110albii 1821 . . . 4 (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤𝑢𝑣(𝑤𝐴𝑣𝑣 = 𝑢))
12 nfcv 2979 . . . . . . 7 𝑥𝑤
13 dffun6f.1 . . . . . . 7 𝑥𝐴
14 nfcv 2979 . . . . . . 7 𝑥𝑦
1512, 13, 14nfbr 5089 . . . . . 6 𝑥 𝑤𝐴𝑦
1615nfmov 2643 . . . . 5 𝑥∃*𝑦 𝑤𝐴𝑦
17 nfv 1915 . . . . 5 𝑤∃*𝑦 𝑥𝐴𝑦
18 breq1 5045 . . . . . 6 (𝑤 = 𝑥 → (𝑤𝐴𝑦𝑥𝐴𝑦))
1918mobidv 2632 . . . . 5 (𝑤 = 𝑥 → (∃*𝑦 𝑤𝐴𝑦 ↔ ∃*𝑦 𝑥𝐴𝑦))
2016, 17, 19cbvalv1 2362 . . . 4 (∀𝑤∃*𝑦 𝑤𝐴𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
219, 11, 203bitr3ri 305 . . 3 (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑤𝑢𝑣(𝑤𝐴𝑣𝑣 = 𝑢))
2221anbi2i 625 . 2 ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑤𝑢𝑣(𝑤𝐴𝑣𝑣 = 𝑢)))
231, 22bitr4i 281 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536  wex 1781  ∃*wmo 2620  wnfc 2960   class class class wbr 5042  Rel wrel 5537  Fun wfun 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-id 5437  df-cnv 5540  df-co 5541  df-fun 6336
This theorem is referenced by:  dffun6  6349  funopab  6369  funcnvmpt  30420  dffun3f  45151
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