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Theorem dffun6f 6579
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 6575 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑤𝑢𝑣(𝑤𝐴𝑣𝑣 = 𝑢)))
2 nfcv 2905 . . . . . . 7 𝑦𝑤
3 dffun6f.2 . . . . . . 7 𝑦𝐴
4 nfcv 2905 . . . . . . 7 𝑦𝑣
52, 3, 4nfbr 5190 . . . . . 6 𝑦 𝑤𝐴𝑣
6 nfv 1914 . . . . . 6 𝑣 𝑤𝐴𝑦
7 breq2 5147 . . . . . 6 (𝑣 = 𝑦 → (𝑤𝐴𝑣𝑤𝐴𝑦))
85, 6, 7cbvmow 2603 . . . . 5 (∃*𝑣 𝑤𝐴𝑣 ↔ ∃*𝑦 𝑤𝐴𝑦)
98albii 1819 . . . 4 (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃*𝑦 𝑤𝐴𝑦)
10 df-mo 2540 . . . . 5 (∃*𝑣 𝑤𝐴𝑣 ↔ ∃𝑢𝑣(𝑤𝐴𝑣𝑣 = 𝑢))
1110albii 1819 . . . 4 (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤𝑢𝑣(𝑤𝐴𝑣𝑣 = 𝑢))
12 nfcv 2905 . . . . . . 7 𝑥𝑤
13 dffun6f.1 . . . . . . 7 𝑥𝐴
14 nfcv 2905 . . . . . . 7 𝑥𝑦
1512, 13, 14nfbr 5190 . . . . . 6 𝑥 𝑤𝐴𝑦
1615nfmov 2560 . . . . 5 𝑥∃*𝑦 𝑤𝐴𝑦
17 nfv 1914 . . . . 5 𝑤∃*𝑦 𝑥𝐴𝑦
18 breq1 5146 . . . . . 6 (𝑤 = 𝑥 → (𝑤𝐴𝑦𝑥𝐴𝑦))
1918mobidv 2549 . . . . 5 (𝑤 = 𝑥 → (∃*𝑦 𝑤𝐴𝑦 ↔ ∃*𝑦 𝑥𝐴𝑦))
2016, 17, 19cbvalv1 2343 . . . 4 (∀𝑤∃*𝑦 𝑤𝐴𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
219, 11, 203bitr3ri 302 . . 3 (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑤𝑢𝑣(𝑤𝐴𝑣𝑣 = 𝑢))
2221anbi2i 623 . 2 ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑤𝑢𝑣(𝑤𝐴𝑣𝑣 = 𝑢)))
231, 22bitr4i 278 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538  wex 1779  ∃*wmo 2538  wnfc 2890   class class class wbr 5143  Rel wrel 5690  Fun wfun 6555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-fun 6563
This theorem is referenced by:  dffun6OLD  6580  funopab  6601  funcnvmpt  32677  dffun3f  49201
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