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Theorem funcnv3 6628
Description: A condition showing a class is single-rooted. (See funcnv 6627). (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 5895 . . . . . 6 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
21eqabri 2873 . . . . 5 (𝑦 ∈ ran 𝐴 ↔ ∃𝑥 𝑥𝐴𝑦)
32biimpi 215 . . . 4 (𝑦 ∈ ran 𝐴 → ∃𝑥 𝑥𝐴𝑦)
43biantrurd 531 . . 3 (𝑦 ∈ ran 𝐴 → (∃*𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦)))
54ralbiia 3088 . 2 (∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦 ↔ ∀𝑦 ∈ ran 𝐴(∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
6 funcnv 6627 . 2 (Fun 𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦)
7 df-reu 3375 . . . 4 (∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ ∃!𝑥(𝑥 ∈ dom 𝐴𝑥𝐴𝑦))
8 vex 3477 . . . . . . 7 𝑥 ∈ V
9 vex 3477 . . . . . . 7 𝑦 ∈ V
108, 9breldm 5915 . . . . . 6 (𝑥𝐴𝑦𝑥 ∈ dom 𝐴)
1110pm4.71ri 559 . . . . 5 (𝑥𝐴𝑦 ↔ (𝑥 ∈ dom 𝐴𝑥𝐴𝑦))
1211eubii 2574 . . . 4 (∃!𝑥 𝑥𝐴𝑦 ↔ ∃!𝑥(𝑥 ∈ dom 𝐴𝑥𝐴𝑦))
13 df-eu 2558 . . . 4 (∃!𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
147, 12, 133bitr2i 298 . . 3 (∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
1514ralbii 3090 . 2 (∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ ∀𝑦 ∈ ran 𝐴(∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
165, 6, 153bitr4i 302 1 (Fun 𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394  wex 1773  wcel 2098  ∃*wmo 2527  ∃!weu 2557  wral 3058  ∃!wreu 3372   class class class wbr 5152  ccnv 5681  dom cdm 5682  ran crn 5683  Fun wfun 6547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-fun 6555
This theorem is referenced by:  cantnf2  42785
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