Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  elghomlem1 Structured version   Unicode version

Theorem elghomlem1 21949
 Description: Lemma for elghom 21951. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
Hypothesis
Ref Expression
elghomlem1.1
Assertion
Ref Expression
elghomlem1 GrpOpHom
Distinct variable groups:   ,,,   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem elghomlem1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnexg 5131 . . 3
2 rnexg 5131 . . 3
3 elghomlem1.1 . . . 4
43fabexg 5624 . . 3
51, 2, 4syl2an 464 . 2
6 rneq 5095 . . . . . 6
76feq2d 5581 . . . . 5
8 oveq 6087 . . . . . . . . 9
98fveq2d 5732 . . . . . . . 8
109eqeq2d 2447 . . . . . . 7
116, 10raleqbidv 2916 . . . . . 6
126, 11raleqbidv 2916 . . . . 5
137, 12anbi12d 692 . . . 4
1413abbidv 2550 . . 3
15 rneq 5095 . . . . . . 7
16 feq3 5578 . . . . . . 7
1715, 16syl 16 . . . . . 6
18 oveq 6087 . . . . . . . 8
1918eqeq1d 2444 . . . . . . 7
20192ralbidv 2747 . . . . . 6
2117, 20anbi12d 692 . . . . 5
2221abbidv 2550 . . . 4
2322, 3syl6eqr 2486 . . 3
24 df-ghom 21946 . . 3 GrpOpHom
2514, 23, 24ovmpt2g 6208 . 2 GrpOpHom
265, 25mpd3an3 1280 1 GrpOpHom
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  cab 2422  wral 2705  cvv 2956   crn 4879  wf 5450  cfv 5454  (class class class)co 6081  cgr 21774   GrpOpHom cghom 21945 This theorem is referenced by:  elghomlem2  21950 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-ghom 21946
 Copyright terms: Public domain W3C validator