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Theorem wdomen2 9029
Description: Equality-like theorem for equinumerosity and weak dominance. (Contributed by Mario Carneiro, 18-May-2015.)
Assertion
Ref Expression
wdomen2 (𝐴𝐵 → (𝐶* 𝐴𝐶* 𝐵))

Proof of Theorem wdomen2
StepHypRef Expression
1 id 22 . . 3 (𝐶* 𝐴𝐶* 𝐴)
2 endom 8524 . . . 4 (𝐴𝐵𝐴𝐵)
3 domwdom 9026 . . . 4 (𝐴𝐵𝐴* 𝐵)
42, 3syl 17 . . 3 (𝐴𝐵𝐴* 𝐵)
5 wdomtr 9027 . . 3 ((𝐶* 𝐴𝐴* 𝐵) → 𝐶* 𝐵)
61, 4, 5syl2anr 596 . 2 ((𝐴𝐵𝐶* 𝐴) → 𝐶* 𝐵)
7 id 22 . . 3 (𝐶* 𝐵𝐶* 𝐵)
8 ensym 8546 . . . 4 (𝐴𝐵𝐵𝐴)
9 endom 8524 . . . 4 (𝐵𝐴𝐵𝐴)
10 domwdom 9026 . . . 4 (𝐵𝐴𝐵* 𝐴)
118, 9, 103syl 18 . . 3 (𝐴𝐵𝐵* 𝐴)
12 wdomtr 9027 . . 3 ((𝐶* 𝐵𝐵* 𝐴) → 𝐶* 𝐴)
137, 11, 12syl2anr 596 . 2 ((𝐴𝐵𝐶* 𝐵) → 𝐶* 𝐴)
146, 13impbida 797 1 (𝐴𝐵 → (𝐶* 𝐴𝐶* 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   class class class wbr 5057  cen 8494  cdom 8495  * cwdom 9009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-wdom 9011
This theorem is referenced by: (None)
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