MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wdomen2 Structured version   Visualization version   GIF version

Theorem wdomen2 8365
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 7868 . . . 4 (𝐴𝐵𝐴𝐵)
3 domwdom 8362 . . . 4 (𝐴𝐵𝐴* 𝐵)
42, 3syl 17 . . 3 (𝐴𝐵𝐴* 𝐵)
5 wdomtr 8363 . . 3 ((𝐶* 𝐴𝐴* 𝐵) → 𝐶* 𝐵)
61, 4, 5syl2anr 494 . 2 ((𝐴𝐵𝐶* 𝐴) → 𝐶* 𝐵)
7 id 22 . . 3 (𝐶* 𝐵𝐶* 𝐵)
8 ensym 7891 . . . 4 (𝐴𝐵𝐵𝐴)
9 endom 7868 . . . 4 (𝐵𝐴𝐵𝐴)
10 domwdom 8362 . . . 4 (𝐵𝐴𝐵* 𝐴)
118, 9, 103syl 18 . . 3 (𝐴𝐵𝐵* 𝐴)
12 wdomtr 8363 . . 3 ((𝐶* 𝐵𝐵* 𝐴) → 𝐶* 𝐴)
137, 11, 12syl2anr 494 . 2 ((𝐴𝐵𝐶* 𝐵) → 𝐶* 𝐴)
146, 13impbida 873 1 (𝐴𝐵 → (𝐶* 𝐴𝐶* 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195   class class class wbr 4583  cen 7838  cdom 7839  * cwdom 8345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-wdom 8347
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator