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Theorem dffun4f 4918
 Description: Definition of function like dffun4 4913 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 4915 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
4 nfcv 2178 . . . . . . 7 𝑦𝑥
5 nfcv 2178 . . . . . . 7 𝑦𝑤
64, 2, 5nfbr 3808 . . . . . 6 𝑦 𝑥𝐴𝑤
7 breq2 3768 . . . . . 6 (𝑦 = 𝑤 → (𝑥𝐴𝑦𝑥𝐴𝑤))
86, 7mo4f 1960 . . . . 5 (∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑦𝑤((𝑥𝐴𝑦𝑥𝐴𝑤) → 𝑦 = 𝑤))
9 nfv 1421 . . . . . . 7 𝑤((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)
10 nfcv 2178 . . . . . . . . . 10 𝑧𝑥
11 dffun4f.3 . . . . . . . . . 10 𝑧𝐴
12 nfcv 2178 . . . . . . . . . 10 𝑧𝑦
1310, 11, 12nfbr 3808 . . . . . . . . 9 𝑧 𝑥𝐴𝑦
14 nfcv 2178 . . . . . . . . . 10 𝑧𝑤
1510, 11, 14nfbr 3808 . . . . . . . . 9 𝑧 𝑥𝐴𝑤
1613, 15nfan 1457 . . . . . . . 8 𝑧(𝑥𝐴𝑦𝑥𝐴𝑤)
17 nfv 1421 . . . . . . . 8 𝑧 𝑦 = 𝑤
1816, 17nfim 1464 . . . . . . 7 𝑧((𝑥𝐴𝑦𝑥𝐴𝑤) → 𝑦 = 𝑤)
19 breq2 3768 . . . . . . . . 9 (𝑧 = 𝑤 → (𝑥𝐴𝑧𝑥𝐴𝑤))
2019anbi2d 437 . . . . . . . 8 (𝑧 = 𝑤 → ((𝑥𝐴𝑦𝑥𝐴𝑧) ↔ (𝑥𝐴𝑦𝑥𝐴𝑤)))
21 equequ2 1599 . . . . . . . 8 (𝑧 = 𝑤 → (𝑦 = 𝑧𝑦 = 𝑤))
2220, 21imbi12d 223 . . . . . . 7 (𝑧 = 𝑤 → (((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ((𝑥𝐴𝑦𝑥𝐴𝑤) → 𝑦 = 𝑤)))
239, 18, 22cbval 1637 . . . . . 6 (∀𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑤((𝑥𝐴𝑦𝑥𝐴𝑤) → 𝑦 = 𝑤))
2423albii 1359 . . . . 5 (∀𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦𝑤((𝑥𝐴𝑦𝑥𝐴𝑤) → 𝑦 = 𝑤))
258, 24bitr4i 176 . . . 4 (∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
2625albii 1359 . . 3 (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
2726anbi2i 430 . 2 ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
28 df-br 3765 . . . . . . 7 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
29 df-br 3765 . . . . . . 7 (𝑥𝐴𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐴)
3028, 29anbi12i 433 . . . . . 6 ((𝑥𝐴𝑦𝑥𝐴𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
3130imbi1i 227 . . . . 5 (((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧))
32312albii 1360 . . . 4 (∀𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧))
3332albii 1359 . . 3 (∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧))
3433anbi2i 430 . 2 ((Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)) ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧)))
353, 27, 343bitri 195 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 97   ↔ wb 98  ∀wal 1241   ∈ wcel 1393  ∃*wmo 1901  Ⅎwnfc 2165  ⟨cop 3378   class class class wbr 3764  Rel wrel 4350  Fun wfun 4896 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-cnv 4353  df-co 4354  df-fun 4904 This theorem is referenced by: (None)
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