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Theorem funcnv3 5917
Description: A condition showing a class is single-rooted. (See funcnv 5916). (Contributed by NM, 26-May-2006.)
Assertion
Ref Expression
funcnv3 (Fun 𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦)
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem funcnv3
StepHypRef Expression
1 dfrn2 5271 . . . . . 6 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
21abeq2i 2732 . . . . 5 (𝑦 ∈ ran 𝐴 ↔ ∃𝑥 𝑥𝐴𝑦)
32biimpi 206 . . . 4 (𝑦 ∈ ran 𝐴 → ∃𝑥 𝑥𝐴𝑦)
43biantrurd 529 . . 3 (𝑦 ∈ ran 𝐴 → (∃*𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦)))
54ralbiia 2973 . 2 (∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦 ↔ ∀𝑦 ∈ ran 𝐴(∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
6 funcnv 5916 . 2 (Fun 𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦)
7 df-reu 2914 . . . 4 (∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ ∃!𝑥(𝑥 ∈ dom 𝐴𝑥𝐴𝑦))
8 vex 3189 . . . . . . 7 𝑥 ∈ V
9 vex 3189 . . . . . . 7 𝑦 ∈ V
108, 9breldm 5289 . . . . . 6 (𝑥𝐴𝑦𝑥 ∈ dom 𝐴)
1110pm4.71ri 664 . . . . 5 (𝑥𝐴𝑦 ↔ (𝑥 ∈ dom 𝐴𝑥𝐴𝑦))
1211eubii 2491 . . . 4 (∃!𝑥 𝑥𝐴𝑦 ↔ ∃!𝑥(𝑥 ∈ dom 𝐴𝑥𝐴𝑦))
13 eu5 2495 . . . 4 (∃!𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
147, 12, 133bitr2i 288 . . 3 (∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
1514ralbii 2974 . 2 (∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ ∀𝑦 ∈ ran 𝐴(∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
165, 6, 153bitr4i 292 1 (Fun 𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wex 1701  wcel 1987  ∃!weu 2469  ∃*wmo 2470  wral 2907  ∃!wreu 2909   class class class wbr 4613  ccnv 5073  dom cdm 5074  ran crn 5075  Fun wfun 5841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-reu 2914  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-fun 5849
This theorem is referenced by: (None)
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