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Theorem dffun4f 4945
Description: Definition of function like dffun4 4940 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 4942 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
4 nfcv 2194 . . . . . . 7 𝑦𝑥
5 nfcv 2194 . . . . . . 7 𝑦𝑤
64, 2, 5nfbr 3835 . . . . . 6 𝑦 𝑥𝐴𝑤
7 breq2 3795 . . . . . 6 (𝑦 = 𝑤 → (𝑥𝐴𝑦𝑥𝐴𝑤))
86, 7mo4f 1976 . . . . 5 (∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑦𝑤((𝑥𝐴𝑦𝑥𝐴𝑤) → 𝑦 = 𝑤))
9 nfv 1437 . . . . . . 7 𝑤((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)
10 nfcv 2194 . . . . . . . . . 10 𝑧𝑥
11 dffun4f.3 . . . . . . . . . 10 𝑧𝐴
12 nfcv 2194 . . . . . . . . . 10 𝑧𝑦
1310, 11, 12nfbr 3835 . . . . . . . . 9 𝑧 𝑥𝐴𝑦
14 nfcv 2194 . . . . . . . . . 10 𝑧𝑤
1510, 11, 14nfbr 3835 . . . . . . . . 9 𝑧 𝑥𝐴𝑤
1613, 15nfan 1473 . . . . . . . 8 𝑧(𝑥𝐴𝑦𝑥𝐴𝑤)
17 nfv 1437 . . . . . . . 8 𝑧 𝑦 = 𝑤
1816, 17nfim 1480 . . . . . . 7 𝑧((𝑥𝐴𝑦𝑥𝐴𝑤) → 𝑦 = 𝑤)
19 breq2 3795 . . . . . . . . 9 (𝑧 = 𝑤 → (𝑥𝐴𝑧𝑥𝐴𝑤))
2019anbi2d 445 . . . . . . . 8 (𝑧 = 𝑤 → ((𝑥𝐴𝑦𝑥𝐴𝑧) ↔ (𝑥𝐴𝑦𝑥𝐴𝑤)))
21 equequ2 1615 . . . . . . . 8 (𝑧 = 𝑤 → (𝑦 = 𝑧𝑦 = 𝑤))
2220, 21imbi12d 227 . . . . . . 7 (𝑧 = 𝑤 → (((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ((𝑥𝐴𝑦𝑥𝐴𝑤) → 𝑦 = 𝑤)))
239, 18, 22cbval 1653 . . . . . 6 (∀𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑤((𝑥𝐴𝑦𝑥𝐴𝑤) → 𝑦 = 𝑤))
2423albii 1375 . . . . 5 (∀𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦𝑤((𝑥𝐴𝑦𝑥𝐴𝑤) → 𝑦 = 𝑤))
258, 24bitr4i 180 . . . 4 (∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
2625albii 1375 . . 3 (∀𝑥∃*𝑦 𝑥𝐴𝑦 ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
2726anbi2i 438 . 2 ((Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
28 df-br 3792 . . . . . . 7 (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
29 df-br 3792 . . . . . . 7 (𝑥𝐴𝑧 ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐴)
3028, 29anbi12i 441 . . . . . 6 ((𝑥𝐴𝑦𝑥𝐴𝑧) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
3130imbi1i 231 . . . . 5 (((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧))
32312albii 1376 . . . 4 (∀𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧))
3332albii 1375 . . 3 (∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧))
3433anbi2i 438 . 2 ((Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)) ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧)))
353, 27, 343bitri 199 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 101  wb 102  wal 1257  wcel 1409  ∃*wmo 1917  wnfc 2181  cop 3405   class class class wbr 3791  Rel wrel 4377  Fun wfun 4923
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 3902  ax-pow 3954  ax-pr 3971
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 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-id 4057  df-cnv 4380  df-co 4381  df-fun 4931
This theorem is referenced by: (None)
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