MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wemaplem3 Unicode version

Theorem wemaplem3 7994
Description: Lemma for wemapso 7997. Transitivity. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Hypotheses
Ref Expression
wemapso.t
wemaplem2.a
wemaplem2.p
wemaplem2.x
wemaplem2.q
wemaplem2.r
wemaplem2.s
wemaplem3.px
wemaplem3.xq
Assertion
Ref Expression
wemaplem3
Distinct variable groups:   ,   , , , ,   , , , ,   ,P, , ,   ,Q, , ,   , , , ,   ,S, , ,

Proof of Theorem wemaplem3
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wemaplem3.px . . 3
2 wemaplem2.p . . . 4
3 wemaplem2.x . . . 4
4 wemapso.t . . . . 5
54wemaplem1 7992 . . . 4
62, 3, 5syl2anc 661 . . 3
71, 6mpbid 210 . 2
8 wemaplem3.xq . . 3
9 wemaplem2.q . . . 4
104wemaplem1 7992 . . . 4
113, 9, 10syl2anc 661 . . 3
128, 11mpbid 210 . 2
13 wemaplem2.a . . . . . 6
1413ad2antrr 725 . . . . 5
152ad2antrr 725 . . . . 5
163ad2antrr 725 . . . . 5
179ad2antrr 725 . . . . 5
18 wemaplem2.r . . . . . 6
1918ad2antrr 725 . . . . 5
20 wemaplem2.s . . . . . 6
2120ad2antrr 725 . . . . 5
22 simplrl 761 . . . . 5
23 simp2rl 1065 . . . . . 6
24233expa 1196 . . . . 5
25 simprr 757 . . . . . 6
2625ad2antlr 726 . . . . 5
27 simprl 756 . . . . 5
28 simprrl 765 . . . . 5
29 simprrr 766 . . . . 5
304, 14, 15, 16, 17, 19, 21, 22, 24, 26, 27, 28, 29wemaplem2 7993 . . . 4
3130rexlimdvaa 2950 . . 3
3231rexlimdvaa 2950 . 2
337, 12, 32mp2d 45 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  A.wral 2807  E.wrex 2808   cvv 3109   class class class wbr 4452  {copab 4509  Powpo 4803  Orwor 4804  `cfv 5593  (class class class)co 6296   cmap 7439
This theorem is referenced by:  wemappo  7995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-po 4805  df-so 4806  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fv 5601  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6800  df-2nd 6801  df-map 7441
  Copyright terms: Public domain W3C validator