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Theorem dffun4f 5139
Description: Definition of function like dffun4 5134 but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 17-Mar-2019.)
Hypotheses
Ref Expression
dffun4f.1 𝑥𝐴
dffun4f.2 𝑦𝐴
dffun4f.3 𝑧𝐴
Assertion
Ref Expression
dffun4f (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧)))
Distinct variable group:   𝑥,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)

Proof of Theorem dffun4f
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 dffun4f.1 . . 3 𝑥𝐴
2 dffun4f.2 . . 3 𝑦𝐴
31, 2dffun6f 5136 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
4 nfcv 2281 . . . . . . 7 𝑦𝑥
5 nfcv 2281 . . . . . . 7 𝑦𝑤
64, 2, 5nfbr 3974 . . . . . 6 𝑦 𝑥𝐴𝑤
7 breq2 3933 . . . . . 6 (𝑦 = 𝑤 → (𝑥𝐴𝑦𝑥𝐴𝑤))
86, 7mo4f 2059 . . . . 5 (∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑦𝑤((𝑥𝐴𝑦𝑥𝐴𝑤) → 𝑦 = 𝑤))
9 nfv 1508 . . . . . . 7 𝑤((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)
10 nfcv 2281 . . . . . . . . . 10 𝑧𝑥
11 dffun4f.3 . . . . . . . . . 10 𝑧𝐴
12 nfcv 2281 . . . . . . . . . 10 𝑧𝑦
1310, 11, 12nfbr 3974 . . . . . . . . 9 𝑧 𝑥𝐴𝑦
14 nfcv 2281 . . . . . . . . . 10 𝑧𝑤
1510, 11, 14nfbr 3974 . . . . . . . . 9 𝑧 𝑥𝐴𝑤
1613, 15nfan 1544 . . . . . . . 8 𝑧(𝑥𝐴𝑦𝑥𝐴𝑤)
17 nfv 1508 . . . . . . . 8 𝑧 𝑦 = 𝑤
1816, 17nfim 1551 . . . . . . 7 𝑧((𝑥𝐴𝑦𝑥𝐴𝑤) → 𝑦 = 𝑤)
19 breq2 3933 . . . . . . . . 9 (𝑧 = 𝑤 → (𝑥𝐴𝑧𝑥𝐴𝑤))
2019anbi2d 459 . . . . . . . 8 (𝑧 = 𝑤 → ((𝑥𝐴𝑦𝑥𝐴𝑧) ↔ (𝑥𝐴𝑦𝑥𝐴𝑤)))
21 equequ2 1689 . . . . . . . 8 (𝑧 = 𝑤 → (𝑦 = 𝑧𝑦 = 𝑤))
2220, 21imbi12d 233 . . . . . . 7 (𝑧 = 𝑤 → (((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ((𝑥𝐴𝑦𝑥𝐴𝑤) → 𝑦 = 𝑤)))
239, 18, 22cbval 1727 . . . . . 6 (∀𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑤((𝑥𝐴𝑦𝑥𝐴𝑤) → 𝑦 = 𝑤))
2423albii 1446 . . . . 5 (∀𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦𝑤((𝑥𝐴𝑦𝑥𝐴𝑤) → 𝑦 = 𝑤))
258, 24bitr4i 186 . . . 4 (∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
2625albii 1446 . . 3 (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
2726anbi2i 452 . 2 ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
28 df-br 3930 . . . . . . 7 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
29 df-br 3930 . . . . . . 7 (𝑥𝐴𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐴)
3028, 29anbi12i 455 . . . . . 6 ((𝑥𝐴𝑦𝑥𝐴𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
3130imbi1i 237 . . . . 5 (((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧))
32312albii 1447 . . . 4 (∀𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧))
3332albii 1446 . . 3 (∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧))
3433anbi2i 452 . 2 ((Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)) ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧)))
353, 27, 343bitri 205 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1329  wcel 1480  ∃*wmo 2000  wnfc 2268  cop 3530   class class class wbr 3929  Rel wrel 4544  Fun wfun 5117
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-id 4215  df-cnv 4547  df-co 4548  df-fun 5125
This theorem is referenced by: (None)
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