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Theorem wdomen2 8442
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 7942 . . . 4 (𝐴𝐵𝐴𝐵)
3 domwdom 8439 . . . 4 (𝐴𝐵𝐴* 𝐵)
42, 3syl 17 . . 3 (𝐴𝐵𝐴* 𝐵)
5 wdomtr 8440 . . 3 ((𝐶* 𝐴𝐴* 𝐵) → 𝐶* 𝐵)
61, 4, 5syl2anr 495 . 2 ((𝐴𝐵𝐶* 𝐴) → 𝐶* 𝐵)
7 id 22 . . 3 (𝐶* 𝐵𝐶* 𝐵)
8 ensym 7965 . . . 4 (𝐴𝐵𝐵𝐴)
9 endom 7942 . . . 4 (𝐵𝐴𝐵𝐴)
10 domwdom 8439 . . . 4 (𝐵𝐴𝐵* 𝐴)
118, 9, 103syl 18 . . 3 (𝐴𝐵𝐵* 𝐴)
12 wdomtr 8440 . . 3 ((𝐶* 𝐵𝐵* 𝐴) → 𝐶* 𝐴)
137, 11, 12syl2anr 495 . 2 ((𝐴𝐵𝐶* 𝐵) → 𝐶* 𝐴)
146, 13impbida 876 1 (𝐴𝐵 → (𝐶* 𝐴𝐶* 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   class class class wbr 4623  cen 7912  cdom 7913  * cwdom 8422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-wdom 8424
This theorem is referenced by: (None)
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