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Theorem dffun6f 6045
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 6042 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑤𝑢𝑣(𝑤𝐴𝑣𝑣 = 𝑢)))
2 nfcv 2913 . . . . . . 7 𝑦𝑤
3 dffun6f.2 . . . . . . 7 𝑦𝐴
4 nfcv 2913 . . . . . . 7 𝑦𝑣
52, 3, 4nfbr 4833 . . . . . 6 𝑦 𝑤𝐴𝑣
6 nfv 1995 . . . . . 6 𝑣 𝑤𝐴𝑦
7 breq2 4790 . . . . . 6 (𝑣 = 𝑦 → (𝑤𝐴𝑣𝑤𝐴𝑦))
85, 6, 7cbvmo 2655 . . . . 5 (∃*𝑣 𝑤𝐴𝑣 ↔ ∃*𝑦 𝑤𝐴𝑦)
98albii 1895 . . . 4 (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃*𝑦 𝑤𝐴𝑦)
10 mo2v 2625 . . . . 5 (∃*𝑣 𝑤𝐴𝑣 ↔ ∃𝑢𝑣(𝑤𝐴𝑣𝑣 = 𝑢))
1110albii 1895 . . . 4 (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤𝑢𝑣(𝑤𝐴𝑣𝑣 = 𝑢))
12 nfcv 2913 . . . . . . 7 𝑥𝑤
13 dffun6f.1 . . . . . . 7 𝑥𝐴
14 nfcv 2913 . . . . . . 7 𝑥𝑦
1512, 13, 14nfbr 4833 . . . . . 6 𝑥 𝑤𝐴𝑦
1615nfmo 2635 . . . . 5 𝑥∃*𝑦 𝑤𝐴𝑦
17 nfv 1995 . . . . 5 𝑤∃*𝑦 𝑥𝐴𝑦
18 breq1 4789 . . . . . 6 (𝑤 = 𝑥 → (𝑤𝐴𝑦𝑥𝐴𝑦))
1918mobidv 2639 . . . . 5 (𝑤 = 𝑥 → (∃*𝑦 𝑤𝐴𝑦 ↔ ∃*𝑦 𝑥𝐴𝑦))
2016, 17, 19cbval 2432 . . . 4 (∀𝑤∃*𝑦 𝑤𝐴𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
219, 11, 203bitr3ri 291 . . 3 (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑤𝑢𝑣(𝑤𝐴𝑣𝑣 = 𝑢))
2221anbi2i 609 . 2 ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑤𝑢𝑣(𝑤𝐴𝑣𝑣 = 𝑢)))
231, 22bitr4i 267 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wal 1629  wex 1852  ∃*wmo 2619  wnfc 2900   class class class wbr 4786  Rel wrel 5254  Fun wfun 6025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-id 5157  df-cnv 5257  df-co 5258  df-fun 6033
This theorem is referenced by:  dffun6  6046  funopab  6066  funcnvmptOLD  29807  funcnvmpt  29808  dffun3f  42957
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