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Theorem funcnv3 6137
Description: A condition showing a class is single-rooted. (See funcnv 6136). (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 5479 . . . . . 6 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
21abeq2i 2878 . . . . 5 (𝑦 ∈ ran 𝐴 ↔ ∃𝑥 𝑥𝐴𝑦)
32biimpi 207 . . . 4 (𝑦 ∈ ran 𝐴 → ∃𝑥 𝑥𝐴𝑦)
43biantrurd 528 . . 3 (𝑦 ∈ ran 𝐴 → (∃*𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦)))
54ralbiia 3126 . 2 (∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦 ↔ ∀𝑦 ∈ ran 𝐴(∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
6 funcnv 6136 . 2 (Fun 𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦)
7 df-reu 3062 . . . 4 (∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ ∃!𝑥(𝑥 ∈ dom 𝐴𝑥𝐴𝑦))
8 vex 3353 . . . . . . 7 𝑥 ∈ V
9 vex 3353 . . . . . . 7 𝑦 ∈ V
108, 9breldm 5497 . . . . . 6 (𝑥𝐴𝑦𝑥 ∈ dom 𝐴)
1110pm4.71ri 556 . . . . 5 (𝑥𝐴𝑦 ↔ (𝑥 ∈ dom 𝐴𝑥𝐴𝑦))
1211eubii 2584 . . . 4 (∃!𝑥 𝑥𝐴𝑦 ↔ ∃!𝑥(𝑥 ∈ dom 𝐴𝑥𝐴𝑦))
13 df-eu 2582 . . . 4 (∃!𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
147, 12, 133bitr2i 290 . . 3 (∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
1514ralbii 3127 . 2 (∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ ∀𝑦 ∈ ran 𝐴(∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
165, 6, 153bitr4i 294 1 (Fun 𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  wex 1874  wcel 2155  ∃*wmo 2563  ∃!weu 2581  wral 3055  ∃!wreu 3057   class class class wbr 4809  ccnv 5276  dom cdm 5277  ran crn 5278  Fun wfun 6062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-reu 3062  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-fun 6070
This theorem is referenced by: (None)
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