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Theorem dffun6f 5148
 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 5145 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑤𝑣𝑢((𝑤𝐴𝑣𝑤𝐴𝑢) → 𝑣 = 𝑢)))
2 nfcv 2283 . . . . . . 7 𝑦𝑤
3 dffun6f.2 . . . . . . 7 𝑦𝐴
4 nfcv 2283 . . . . . . 7 𝑦𝑣
52, 3, 4nfbr 3984 . . . . . 6 𝑦 𝑤𝐴𝑣
6 nfv 1505 . . . . . 6 𝑣 𝑤𝐴𝑦
7 breq2 3943 . . . . . 6 (𝑣 = 𝑦 → (𝑤𝐴𝑣𝑤𝐴𝑦))
85, 6, 7cbvmo 2030 . . . . 5 (∃*𝑣 𝑤𝐴𝑣 ↔ ∃*𝑦 𝑤𝐴𝑦)
98albii 1447 . . . 4 (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃*𝑦 𝑤𝐴𝑦)
10 breq2 3943 . . . . . 6 (𝑣 = 𝑢 → (𝑤𝐴𝑣𝑤𝐴𝑢))
1110mo4 2051 . . . . 5 (∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑣𝑢((𝑤𝐴𝑣𝑤𝐴𝑢) → 𝑣 = 𝑢))
1211albii 1447 . . . 4 (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤𝑣𝑢((𝑤𝐴𝑣𝑤𝐴𝑢) → 𝑣 = 𝑢))
13 nfcv 2283 . . . . . . 7 𝑥𝑤
14 dffun6f.1 . . . . . . 7 𝑥𝐴
15 nfcv 2283 . . . . . . 7 𝑥𝑦
1613, 14, 15nfbr 3984 . . . . . 6 𝑥 𝑤𝐴𝑦
1716nfmo 2010 . . . . 5 𝑥∃*𝑦 𝑤𝐴𝑦
18 nfv 1505 . . . . 5 𝑤∃*𝑦 𝑥𝐴𝑦
19 breq1 3942 . . . . . 6 (𝑤 = 𝑥 → (𝑤𝐴𝑦𝑥𝐴𝑦))
2019mobidv 2026 . . . . 5 (𝑤 = 𝑥 → (∃*𝑦 𝑤𝐴𝑦 ↔ ∃*𝑦 𝑥𝐴𝑦))
2117, 18, 20cbval 1723 . . . 4 (∀𝑤∃*𝑦 𝑤𝐴𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
229, 12, 213bitr3ri 210 . . 3 (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑤𝑣𝑢((𝑤𝐴𝑣𝑤𝐴𝑢) → 𝑣 = 𝑢))
2322anbi2i 453 . 2 ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑤𝑣𝑢((𝑤𝐴𝑣𝑤𝐴𝑢) → 𝑣 = 𝑢)))
241, 23bitr4i 186 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1330  ∃*wmo 1991  Ⅎwnfc 2270   class class class wbr 3939  Rel wrel 4556  Fun wfun 5129 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2115  ax-ext 2123  ax-sep 4056  ax-pow 4108  ax-pr 4142 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1732  df-eu 1993  df-mo 1994  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ral 2423  df-v 2693  df-un 3082  df-in 3084  df-ss 3091  df-pw 3519  df-sn 3540  df-pr 3541  df-op 3543  df-br 3940  df-opab 4000  df-id 4226  df-cnv 4559  df-co 4560  df-fun 5137 This theorem is referenced by:  dffun6  5149  dffun4f  5151  funopab  5170
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