Theorem cntzmhm2 18166

Description: Centralizers in a monoid are preserved by monoid homomorphisms. (Contributed by Mario Carneiro, 24-Apr-2016.)

Hypotheses

Ref	Expression
cntzmhm.z	⊢ 𝑍 = (Cntz‘𝐺)
cntzmhm.y	⊢ 𝑌 = (Cntz‘𝐻)

Assertion

Ref	Expression
cntzmhm2	⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → (𝐹 “ 𝑆) ⊆ (𝑌‘(𝐹 “ 𝑇)))

Proof of Theorem cntzmhm2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cntzmhm.z	. . . . 5 ⊢ 𝑍 = (Cntz‘𝐺)
2		cntzmhm.y	. . . . 5 ⊢ 𝑌 = (Cntz‘𝐻)
3	1, 2	cntzmhm 18165	. . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑥 ∈ (𝑍‘𝑇)) → (𝐹‘𝑥) ∈ (𝑌‘(𝐹 “ 𝑇)))
4	3	ralrimiva 3148	. . 3 ⊢ (𝐹 ∈ (𝐺 MndHom 𝐻) → ∀𝑥 ∈ (𝑍‘𝑇)(𝐹‘𝑥) ∈ (𝑌‘(𝐹 “ 𝑇)))
5		ssralv 3885	. . 3 ⊢ (𝑆 ⊆ (𝑍‘𝑇) → (∀𝑥 ∈ (𝑍‘𝑇)(𝐹‘𝑥) ∈ (𝑌‘(𝐹 “ 𝑇)) → ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) ∈ (𝑌‘(𝐹 “ 𝑇))))
6	4, 5	mpan9 502	. 2 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) ∈ (𝑌‘(𝐹 “ 𝑇)))
7		eqid 2778	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
8		eqid 2778	. . . . . 6 ⊢ (Base‘𝐻) = (Base‘𝐻)
9	7, 8	mhmf 17737	. . . . 5 ⊢ (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
10	9	adantr 474	. . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
11	10	ffund 6297	. . 3 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → Fun 𝐹)
12		simpr 479	. . . . 5 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → 𝑆 ⊆ (𝑍‘𝑇))
13	7, 1	cntzssv 18155	. . . . 5 ⊢ (𝑍‘𝑇) ⊆ (Base‘𝐺)
14	12, 13	syl6ss 3833	. . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → 𝑆 ⊆ (Base‘𝐺))
15	10	fdmd 6302	. . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → dom 𝐹 = (Base‘𝐺))
16	14, 15	sseqtr4d 3861	. . 3 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → 𝑆 ⊆ dom 𝐹)
17		funimass4 6509	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑆 ⊆ dom 𝐹) → ((𝐹 “ 𝑆) ⊆ (𝑌‘(𝐹 “ 𝑇)) ↔ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) ∈ (𝑌‘(𝐹 “ 𝑇))))
18	11, 16, 17	syl2anc 579	. 2 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → ((𝐹 “ 𝑆) ⊆ (𝑌‘(𝐹 “ 𝑇)) ↔ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) ∈ (𝑌‘(𝐹 “ 𝑇))))
19	6, 18	mpbird 249	1 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → (𝐹 “ 𝑆) ⊆ (𝑌‘(𝐹 “ 𝑇)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601   ∈ wcel 2107  ∀wral 3090   ⊆ wss 3792  dom cdm 5357   “ cima 5360  Fun wfun 6131  ⟶wf 6133  ‘cfv 6137  (class class class)co 6924  Basecbs 16266   MndHom cmhm 17730  Cntzccntz 18142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-map 8144  df-mhm 17732  df-cntz 18144

This theorem is referenced by:  gsumzmhm  18734  gsumzinv  18742