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Theorem funcnv3 6576
Description: A condition showing a class is single-rooted. (See funcnv 6575). (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 5849 . . . . . 6 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
21eqabi 2882 . . . . 5 (𝑦 ∈ ran 𝐴 ↔ ∃𝑥 𝑥𝐴𝑦)
32biimpi 215 . . . 4 (𝑦 ∈ ran 𝐴 → ∃𝑥 𝑥𝐴𝑦)
43biantrurd 534 . . 3 (𝑦 ∈ ran 𝐴 → (∃*𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦)))
54ralbiia 3095 . 2 (∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦 ↔ ∀𝑦 ∈ ran 𝐴(∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
6 funcnv 6575 . 2 (Fun 𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦)
7 df-reu 3357 . . . 4 (∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ ∃!𝑥(𝑥 ∈ dom 𝐴𝑥𝐴𝑦))
8 vex 3452 . . . . . . 7 𝑥 ∈ V
9 vex 3452 . . . . . . 7 𝑦 ∈ V
108, 9breldm 5869 . . . . . 6 (𝑥𝐴𝑦𝑥 ∈ dom 𝐴)
1110pm4.71ri 562 . . . . 5 (𝑥𝐴𝑦 ↔ (𝑥 ∈ dom 𝐴𝑥𝐴𝑦))
1211eubii 2584 . . . 4 (∃!𝑥 𝑥𝐴𝑦 ↔ ∃!𝑥(𝑥 ∈ dom 𝐴𝑥𝐴𝑦))
13 df-eu 2568 . . . 4 (∃!𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
147, 12, 133bitr2i 299 . . 3 (∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
1514ralbii 3097 . 2 (∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ ∀𝑦 ∈ ran 𝐴(∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
165, 6, 153bitr4i 303 1 (Fun 𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wex 1782  wcel 2107  ∃*wmo 2537  ∃!weu 2567  wral 3065  ∃!wreu 3354   class class class wbr 5110  ccnv 5637  dom cdm 5638  ran crn 5639  Fun wfun 6495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-fun 6503
This theorem is referenced by:  cantnf2  41689
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