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Theorem dffun6f 6553
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 6549 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑤𝑢𝑣(𝑤𝐴𝑣𝑣 = 𝑢)))
2 nfcv 2904 . . . . . . 7 𝑦𝑤
3 dffun6f.2 . . . . . . 7 𝑦𝐴
4 nfcv 2904 . . . . . . 7 𝑦𝑣
52, 3, 4nfbr 5191 . . . . . 6 𝑦 𝑤𝐴𝑣
6 nfv 1918 . . . . . 6 𝑣 𝑤𝐴𝑦
7 breq2 5148 . . . . . 6 (𝑣 = 𝑦 → (𝑤𝐴𝑣𝑤𝐴𝑦))
85, 6, 7cbvmow 2598 . . . . 5 (∃*𝑣 𝑤𝐴𝑣 ↔ ∃*𝑦 𝑤𝐴𝑦)
98albii 1822 . . . 4 (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃*𝑦 𝑤𝐴𝑦)
10 df-mo 2535 . . . . 5 (∃*𝑣 𝑤𝐴𝑣 ↔ ∃𝑢𝑣(𝑤𝐴𝑣𝑣 = 𝑢))
1110albii 1822 . . . 4 (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤𝑢𝑣(𝑤𝐴𝑣𝑣 = 𝑢))
12 nfcv 2904 . . . . . . 7 𝑥𝑤
13 dffun6f.1 . . . . . . 7 𝑥𝐴
14 nfcv 2904 . . . . . . 7 𝑥𝑦
1512, 13, 14nfbr 5191 . . . . . 6 𝑥 𝑤𝐴𝑦
1615nfmov 2555 . . . . 5 𝑥∃*𝑦 𝑤𝐴𝑦
17 nfv 1918 . . . . 5 𝑤∃*𝑦 𝑥𝐴𝑦
18 breq1 5147 . . . . . 6 (𝑤 = 𝑥 → (𝑤𝐴𝑦𝑥𝐴𝑦))
1918mobidv 2544 . . . . 5 (𝑤 = 𝑥 → (∃*𝑦 𝑤𝐴𝑦 ↔ ∃*𝑦 𝑥𝐴𝑦))
2016, 17, 19cbvalv1 2338 . . . 4 (∀𝑤∃*𝑦 𝑤𝐴𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
219, 11, 203bitr3ri 302 . . 3 (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑤𝑢𝑣(𝑤𝐴𝑣𝑣 = 𝑢))
2221anbi2i 624 . 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 205  wa 397  wal 1540  wex 1782  ∃*wmo 2533  wnfc 2884   class class class wbr 5144  Rel wrel 5677  Fun wfun 6529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5145  df-opab 5207  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-fun 6537
This theorem is referenced by:  dffun6OLD  6554  funopab  6575  funcnvmpt  31861  dffun3f  47567
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