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Theorem dffun6f 4943
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 4940 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑤𝑣𝑢((𝑤𝐴𝑣𝑤𝐴𝑢) → 𝑣 = 𝑢)))
2 nfcv 2194 . . . . . . 7 𝑦𝑤
3 dffun6f.2 . . . . . . 7 𝑦𝐴
4 nfcv 2194 . . . . . . 7 𝑦𝑣
52, 3, 4nfbr 3836 . . . . . 6 𝑦 𝑤𝐴𝑣
6 nfv 1437 . . . . . 6 𝑣 𝑤𝐴𝑦
7 breq2 3796 . . . . . 6 (𝑣 = 𝑦 → (𝑤𝐴𝑣𝑤𝐴𝑦))
85, 6, 7cbvmo 1956 . . . . 5 (∃*𝑣 𝑤𝐴𝑣 ↔ ∃*𝑦 𝑤𝐴𝑦)
98albii 1375 . . . 4 (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤∃*𝑦 𝑤𝐴𝑦)
10 breq2 3796 . . . . . 6 (𝑣 = 𝑢 → (𝑤𝐴𝑣𝑤𝐴𝑢))
1110mo4 1977 . . . . 5 (∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑣𝑢((𝑤𝐴𝑣𝑤𝐴𝑢) → 𝑣 = 𝑢))
1211albii 1375 . . . 4 (∀𝑤∃*𝑣 𝑤𝐴𝑣 ↔ ∀𝑤𝑣𝑢((𝑤𝐴𝑣𝑤𝐴𝑢) → 𝑣 = 𝑢))
13 nfcv 2194 . . . . . . 7 𝑥𝑤
14 dffun6f.1 . . . . . . 7 𝑥𝐴
15 nfcv 2194 . . . . . . 7 𝑥𝑦
1613, 14, 15nfbr 3836 . . . . . 6 𝑥 𝑤𝐴𝑦
1716nfmo 1936 . . . . 5 𝑥∃*𝑦 𝑤𝐴𝑦
18 nfv 1437 . . . . 5 𝑤∃*𝑦 𝑥𝐴𝑦
19 breq1 3795 . . . . . 6 (𝑤 = 𝑥 → (𝑤𝐴𝑦𝑥𝐴𝑦))
2019mobidv 1952 . . . . 5 (𝑤 = 𝑥 → (∃*𝑦 𝑤𝐴𝑦 ↔ ∃*𝑦 𝑥𝐴𝑦))
2117, 18, 20cbval 1653 . . . 4 (∀𝑤∃*𝑦 𝑤𝐴𝑦 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
229, 12, 213bitr3ri 204 . . 3 (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑤𝑣𝑢((𝑤𝐴𝑣𝑤𝐴𝑢) → 𝑣 = 𝑢))
2322anbi2i 438 . 2 ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑤𝑣𝑢((𝑤𝐴𝑣𝑤𝐴𝑢) → 𝑣 = 𝑢)))
241, 23bitr4i 180 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257  ∃*wmo 1917  wnfc 2181   class class class wbr 3792  Rel wrel 4378  Fun wfun 4924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-id 4058  df-cnv 4381  df-co 4382  df-fun 4932
This theorem is referenced by:  dffun6  4944  dffun4f  4946  funopab  4963
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