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Theorem dffun2OLDOLD 6575
Description: Obsolete version of dffun2 6573 as of 11-Dec-2024. (Contributed by NM, 29-Dec-1996.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dffun2OLDOLD (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
Distinct variable group:   𝑥,𝐴,𝑦,𝑧

Proof of Theorem dffun2OLDOLD
StepHypRef Expression
1 df-fun 6565 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ))
2 df-id 5583 . . . . . 6 I = {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧}
32sseq2i 4025 . . . . 5 ((𝐴𝐴) ⊆ I ↔ (𝐴𝐴) ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧})
4 df-co 5698 . . . . . 6 (𝐴𝐴) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧)}
54sseq1i 4024 . . . . 5 ((𝐴𝐴) ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧} ↔ {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧)} ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧})
6 ssopab2bw 5557 . . . . 5 ({⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧)} ⊆ {⟨𝑦, 𝑧⟩ ∣ 𝑦 = 𝑧} ↔ ∀𝑦𝑧(∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧))
73, 5, 63bitri 297 . . . 4 ((𝐴𝐴) ⊆ I ↔ ∀𝑦𝑧(∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧))
8 vex 3482 . . . . . . . . . . . 12 𝑦 ∈ V
9 vex 3482 . . . . . . . . . . . 12 𝑥 ∈ V
108, 9brcnv 5896 . . . . . . . . . . 11 (𝑦𝐴𝑥𝑥𝐴𝑦)
1110anbi1i 624 . . . . . . . . . 10 ((𝑦𝐴𝑥𝑥𝐴𝑧) ↔ (𝑥𝐴𝑦𝑥𝐴𝑧))
1211exbii 1845 . . . . . . . . 9 (∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) ↔ ∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧))
1312imbi1i 349 . . . . . . . 8 ((∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ (∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
14 19.23v 1940 . . . . . . . 8 (∀𝑥((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ (∃𝑥(𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
1513, 14bitr4i 278 . . . . . . 7 ((∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
1615albii 1816 . . . . . 6 (∀𝑧(∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑧𝑥((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
17 alcom 2157 . . . . . 6 (∀𝑧𝑥((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
1816, 17bitri 275 . . . . 5 (∀𝑧(∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
1918albii 1816 . . . 4 (∀𝑦𝑧(∃𝑥(𝑦𝐴𝑥𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑦𝑥𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
20 alcom 2157 . . . 4 (∀𝑦𝑥𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧) ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
217, 19, 203bitri 297 . . 3 ((𝐴𝐴) ⊆ I ↔ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧))
2221anbi2i 623 . 2 ((Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ) ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
231, 22bitri 275 1 (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥𝑦𝑧((𝑥𝐴𝑦𝑥𝐴𝑧) → 𝑦 = 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535  wex 1776  wss 3963   class class class wbr 5148  {copab 5210   I cid 5582  ccnv 5688  ccom 5693  Rel wrel 5694  Fun wfun 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-cnv 5697  df-co 5698  df-fun 6565
This theorem is referenced by: (None)
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