Theorem ghgrpilem2 8086

Description: Lemma for ghgrpi 8089.

Hypotheses

Ref	Expression
ghgrpi.1	|- G e. Grp
ghgrpi.2	|- X = ran G
ghgrpi.3	|- F:X-onto->Y
ghgrpi.4	|- Y (_ A
ghgrpi.5	|- O Fn (A X. A)
ghgrpi.6	|- ((x e. X /\ y e. X) -> (F` (xGy)) = ((F` x)O(F` y)))
ghgrpi.7	|- H = (O |` (Y X. Y))
ghgrpilem2.8	|- ((w e. X /\ ph) -> ps)
ghgrpilem2.9	|- ((F` w) = C -> (ps <-> ch))

Assertion

Ref	Expression
ghgrpilem2	|- ((ph /\ C e. Y) -> ch)

Distinct variable groups:   x,F,y   x,G,y   x,H,y   x,O,y   x,X,y   x,Y,y   w,C   w,F   w,X   ch,w   ph,w

Proof of Theorem ghgrpilem2

Step	Hyp	Ref	Expression
1		ghgrpilem2.9	. . . . 5 |- ((F` w) = C -> (ps <-> ch))
2		ghgrpilem2.8	. . . . . 6 |- ((w e. X /\ ph) -> ps)
3	2	ancoms 436	. . . . 5 |- ((ph /\ w e. X) -> ps)
4	1, 3	syl5cbi 209	. . . 4 |- ((ph /\ w e. X) -> ((F` w) = C -> ch))
5	4	r19.23adva 1744	. . 3 |- (ph -> (E.w e. X (F` w) = C -> ch))
6		ghgrpi.3	. . . . . . 7 |- F:X-onto->Y
7		df-fo 3191	. . . . . . 7 |- (F:X-onto->Y <-> (F Fn X /\ ran F = Y))
8	6, 7	mpbi 189	. . . . . 6 |- (F Fn X /\ ran F = Y)
9	8	pm3.27i 324	. . . . 5 |- ran F = Y
10	9	eleq2i 1535	. . . 4 |- (C e. ran F <-> C e. Y)
11	8	pm3.26i 320	. . . . 5 |- F Fn X
12		fvelrnb 3751	. . . . 5 |- (F Fn X -> (C e. ran F <-> E.w e. X (F` w) = C))
13	11, 12	ax-mp 7	. . . 4 |- (C e. ran F <-> E.w e. X (F` w) = C)
14	10, 13	bitr3 175	. . 3 |- (C e. Y <-> E.w e. X (F` w) = C)
15	5, 14	syl5ib 206	. 2 |- (ph -> (C e. Y -> ch))
16	15	imp 350	1 |- ((ph /\ C e. Y) -> ch)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  E.wrex 1643   (_ wss 2043   X. cxp 3163  ran crn 3166   |` cres 3167   Fn wfn 3172  -onto->wfo 3175  ` cfv 3177  (class class class)co 3954  Grpcgr 7983

This theorem is referenced by:  ghgrpilem3 8087  ghgrpilem4 8088  ghgrpi 8089

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fo 3191  df-fv 3193

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