Theorem ghgrpilem2 8071

Description: Lemma for ghgrpi 8074.

Hypotheses

Ref	Expression
ghgrpi.1	⊢ G ∈ Grp
ghgrpi.2	⊢ X = ran G
ghgrpi.3	⊢ F:X–onto→Y
ghgrpi.4	⊢ Y ⊆ A
ghgrpi.5	⊢ O Fn (A × A)
ghgrpi.6	⊢ ((x ∈ X ⋀ y ∈ X) → (F ‘(xGy)) = ((F ‘x)O(F ‘y)))
ghgrpi.7	⊢ H = (O ↾ (Y × Y))
ghgrpilem2.8	⊢ ((w ∈ X ⋀ φ) → ψ)
ghgrpilem2.9	⊢ ((F ‘w) = C → (ψ ↔ χ))

Assertion

Ref	Expression
ghgrpilem2	⊢ ((φ ⋀ C ∈ Y) → χ)

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   χ,w   φ,w

Proof of Theorem ghgrpilem2

Step	Hyp	Ref	Expression
1		ghgrpilem2.9	. . . . 5 ⊢ ((F ‘w) = C → (ψ ↔ χ))
2		ghgrpilem2.8	. . . . . 6 ⊢ ((w ∈ X ⋀ φ) → ψ)
3	2	ancoms 436	. . . . 5 ⊢ ((φ ⋀ w ∈ X) → ψ)
4	1, 3	syl5cbi 209	. . . 4 ⊢ ((φ ⋀ w ∈ X) → ((F ‘w) = C → χ))
5	4	r19.23adva 1739	. . 3 ⊢ (φ → (∃w ∈ X (F ‘w) = C → χ))
6		ghgrpi.3	. . . . . . 7 ⊢ F:X–onto→Y
7		df-fo 3186	. . . . . . 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 1530	. . . 4 ⊢ (C ∈ ran F ↔ C ∈ Y)
11	8	pm3.26i 320	. . . . 5 ⊢ F Fn X
12		fvelrnb 3745	. . . . 5 ⊢ (F Fn X → (C ∈ ran F ↔ ∃w ∈ X (F ‘w) = C))
13	11, 12	ax-mp 7	. . . 4 ⊢ (C ∈ ran F ↔ ∃w ∈ X (F ‘w) = C)
14	10, 13	bitr3 175	. . 3 ⊢ (C ∈ Y ↔ ∃w ∈ X (F ‘w) = C)
15	5, 14	syl5ib 206	. 2 ⊢ (φ → (C ∈ Y → χ))
16	15	imp 350	1 ⊢ ((φ ⋀ C ∈ Y) → χ)

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 953   ∈ wcel 955  ∃wrex 1638   ⊆ wss 2037   × cxp 3158  ran crn 3161   ↾ cres 3162   Fn wfn 3167  –onto→wfo 3170   ‘cfv 3172  (class class class)co 3948  Grpcgr 7967

This theorem is referenced by:  ghgrpilem3 8072  ghgrpilem4 8073  ghgrpi 8074

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fo 3186  df-fv 3188

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