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Theorem dffun6f 5229
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 dffun2 5226 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑤𝑣𝑢((𝑤𝐴𝑣𝑤𝐴𝑢) → 𝑣 = 𝑢)))
2 nfcv 2319 . . . . . . 7 𝑦𝑤
3 dffun6f.2 . . . . . . 7 𝑦𝐴
4 nfcv 2319 . . . . . . 7 𝑦𝑣
52, 3, 4nfbr 4049 . . . . . 6 𝑦 𝑤𝐴𝑣
6 nfv 1528 . . . . . 6 𝑣 𝑤𝐴𝑦
7 breq2 4007 . . . . . 6 (𝑣 = 𝑦 → (𝑤𝐴𝑣𝑤𝐴𝑦))
85, 6, 7cbvmo 2066 . . . . 5 (∃*𝑣 𝑤𝐴𝑣 ↔ ∃*𝑦 𝑤𝐴𝑦)
98albii 1470 . . . 4 (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃*𝑦 𝑤𝐴𝑦)
10 breq2 4007 . . . . . 6 (𝑣 = 𝑢 → (𝑤𝐴𝑣𝑤𝐴𝑢))
1110mo4 2087 . . . . 5 (∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑣𝑢((𝑤𝐴𝑣𝑤𝐴𝑢) → 𝑣 = 𝑢))
1211albii 1470 . . . 4 (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤𝑣𝑢((𝑤𝐴𝑣𝑤𝐴𝑢) → 𝑣 = 𝑢))
13 nfcv 2319 . . . . . . 7 𝑥𝑤
14 dffun6f.1 . . . . . . 7 𝑥𝐴
15 nfcv 2319 . . . . . . 7 𝑥𝑦
1613, 14, 15nfbr 4049 . . . . . 6 𝑥 𝑤𝐴𝑦
1716nfmo 2046 . . . . 5 𝑥∃*𝑦 𝑤𝐴𝑦
18 nfv 1528 . . . . 5 𝑤∃*𝑦 𝑥𝐴𝑦
19 breq1 4006 . . . . . 6 (𝑤 = 𝑥 → (𝑤𝐴𝑦𝑥𝐴𝑦))
2019mobidv 2062 . . . . 5 (𝑤 = 𝑥 → (∃*𝑦 𝑤𝐴𝑦 ↔ ∃*𝑦 𝑥𝐴𝑦))
2117, 18, 20cbval 1754 . . . 4 (∀𝑤∃*𝑦 𝑤𝐴𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
229, 12, 213bitr3ri 211 . . 3 (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑤𝑣𝑢((𝑤𝐴𝑣𝑤𝐴𝑢) → 𝑣 = 𝑢))
2322anbi2i 457 . 2 ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑤𝑣𝑢((𝑤𝐴𝑣𝑤𝐴𝑢) → 𝑣 = 𝑢)))
241, 23bitr4i 187 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1351  ∃*wmo 2027  wnfc 2306   class class class wbr 4003  Rel wrel 4631  Fun wfun 5210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-id 4293  df-cnv 4634  df-co 4635  df-fun 5218
This theorem is referenced by:  dffun6  5230  dffun4f  5232  funopab  5251
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