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Theorem dffun2OLD 6554
Description: Obsolete version of dffun2 6553 as of 29-Dec-2024. (Contributed by NM, 29-Dec-1996.) Avoid ax-10 2137, ax-12 2171. (Revised by SN, 19-Dec-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dffun2OLD (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
Distinct variable group:   𝑥,𝐴,𝑦,𝑧

Proof of Theorem dffun2OLD
StepHypRef Expression
1 df-fun 6545 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ))
2 cotrg 6108 . . . 4 ((𝐴𝐴) ⊆ I ↔ ∀𝑦𝑥𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧))
3 alcom 2156 . . . . 5 (∀𝑦𝑥𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑥𝑦𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧))
4 vex 3478 . . . . . . . . 9 𝑦 ∈ V
5 vex 3478 . . . . . . . . 9 𝑥 ∈ V
64, 5brcnv 5882 . . . . . . . 8 (𝑦𝐴𝑥𝑥𝐴𝑦)
76anbi1i 624 . . . . . . 7 ((𝑦𝐴𝑥𝑥𝐴𝑧) ↔ (𝑥𝐴𝑦𝑥𝐴𝑧))
8 vex 3478 . . . . . . . 8 𝑧 ∈ V
98ideq 5852 . . . . . . 7 (𝑦 I 𝑧𝑦 = 𝑧)
107, 9imbi12i 350 . . . . . 6 (((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
11103albii 1823 . . . . 5 (∀𝑥𝑦𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
123, 11bitri 274 . . . 4 (∀𝑦𝑥𝑧((𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 I 𝑧) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
132, 12bitri 274 . . 3 ((𝐴𝐴) ⊆ I ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
1413anbi2i 623 . 2 ((Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ) ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
151, 14bitri 274 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539  wss 3948   class class class wbr 5148   I cid 5573  ccnv 5675  ccom 5680  Rel wrel 5681  Fun wfun 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-fun 6545
This theorem is referenced by: (None)
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