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Theorem dffun6f 4996
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 4993 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑤𝑣𝑢((𝑤𝐴𝑣𝑤𝐴𝑢) → 𝑣 = 𝑢)))
2 nfcv 2225 . . . . . . 7 𝑦𝑤
3 dffun6f.2 . . . . . . 7 𝑦𝐴
4 nfcv 2225 . . . . . . 7 𝑦𝑣
52, 3, 4nfbr 3866 . . . . . 6 𝑦 𝑤𝐴𝑣
6 nfv 1464 . . . . . 6 𝑣 𝑤𝐴𝑦
7 breq2 3826 . . . . . 6 (𝑣 = 𝑦 → (𝑤𝐴𝑣𝑤𝐴𝑦))
85, 6, 7cbvmo 1985 . . . . 5 (∃*𝑣 𝑤𝐴𝑣 ↔ ∃*𝑦 𝑤𝐴𝑦)
98albii 1402 . . . 4 (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃*𝑦 𝑤𝐴𝑦)
10 breq2 3826 . . . . . 6 (𝑣 = 𝑢 → (𝑤𝐴𝑣𝑤𝐴𝑢))
1110mo4 2006 . . . . 5 (∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑣𝑢((𝑤𝐴𝑣𝑤𝐴𝑢) → 𝑣 = 𝑢))
1211albii 1402 . . . 4 (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤𝑣𝑢((𝑤𝐴𝑣𝑤𝐴𝑢) → 𝑣 = 𝑢))
13 nfcv 2225 . . . . . . 7 𝑥𝑤
14 dffun6f.1 . . . . . . 7 𝑥𝐴
15 nfcv 2225 . . . . . . 7 𝑥𝑦
1613, 14, 15nfbr 3866 . . . . . 6 𝑥 𝑤𝐴𝑦
1716nfmo 1965 . . . . 5 𝑥∃*𝑦 𝑤𝐴𝑦
18 nfv 1464 . . . . 5 𝑤∃*𝑦 𝑥𝐴𝑦
19 breq1 3825 . . . . . 6 (𝑤 = 𝑥 → (𝑤𝐴𝑦𝑥𝐴𝑦))
2019mobidv 1981 . . . . 5 (𝑤 = 𝑥 → (∃*𝑦 𝑤𝐴𝑦 ↔ ∃*𝑦 𝑥𝐴𝑦))
2117, 18, 20cbval 1681 . . . 4 (∀𝑤∃*𝑦 𝑤𝐴𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
229, 12, 213bitr3ri 209 . . 3 (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑤𝑣𝑢((𝑤𝐴𝑣𝑤𝐴𝑢) → 𝑣 = 𝑢))
2322anbi2i 445 . 2 ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑤𝑣𝑢((𝑤𝐴𝑣𝑤𝐴𝑢) → 𝑣 = 𝑢)))
241, 23bitr4i 185 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1285  ∃*wmo 1946  wnfc 2212   class class class wbr 3822  Rel wrel 4418  Fun wfun 4977
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3823  df-opab 3877  df-id 4096  df-cnv 4421  df-co 4422  df-fun 4985
This theorem is referenced by:  dffun6  4997  dffun4f  4999  funopab  5016
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