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Theorem funcnv3 4961
 Description: A condition showing a class is single-rooted. (See funcnv 4960). (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 4523 . . . . . 6 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
21abeq2i 2148 . . . . 5 (𝑦 ∈ ran 𝐴 ↔ ∃𝑥 𝑥𝐴𝑦)
32biimpi 113 . . . 4 (𝑦 ∈ ran 𝐴 → ∃𝑥 𝑥𝐴𝑦)
43biantrurd 289 . . 3 (𝑦 ∈ ran 𝐴 → (∃*𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦)))
54ralbiia 2338 . 2 (∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦 ↔ ∀𝑦 ∈ ran 𝐴(∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
6 funcnv 4960 . 2 (Fun 𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦)
7 df-reu 2313 . . . 4 (∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ ∃!𝑥(𝑥 ∈ dom 𝐴𝑥𝐴𝑦))
8 vex 2560 . . . . . . 7 𝑥 ∈ V
9 vex 2560 . . . . . . 7 𝑦 ∈ V
108, 9breldm 4539 . . . . . 6 (𝑥𝐴𝑦𝑥 ∈ dom 𝐴)
1110pm4.71ri 372 . . . . 5 (𝑥𝐴𝑦 ↔ (𝑥 ∈ dom 𝐴𝑥𝐴𝑦))
1211eubii 1909 . . . 4 (∃!𝑥 𝑥𝐴𝑦 ↔ ∃!𝑥(𝑥 ∈ dom 𝐴𝑥𝐴𝑦))
13 eu5 1947 . . . 4 (∃!𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
147, 12, 133bitr2i 197 . . 3 (∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
1514ralbii 2330 . 2 (∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦 ↔ ∀𝑦 ∈ ran 𝐴(∃𝑥 𝑥𝐴𝑦 ∧ ∃*𝑥 𝑥𝐴𝑦))
165, 6, 153bitr4i 201 1 (Fun 𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  ∃wex 1381   ∈ wcel 1393  ∃!weu 1900  ∃*wmo 1901  ∀wral 2306  ∃!wreu 2308   class class class wbr 3764  ◡ccnv 4344  dom cdm 4345  ran crn 4346  Fun wfun 4896 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-reu 2313  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904 This theorem is referenced by: (None)
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