Theorem cntzmhm2 19221

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 19220	. . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑥 ∈ (𝑍‘𝑇)) → (𝐹‘𝑥) ∈ (𝑌‘(𝐹 “ 𝑇)))
4	3	ralrimiva 3121	. . 3 ⊢ (𝐹 ∈ (𝐺 MndHom 𝐻) → ∀𝑥 ∈ (𝑍‘𝑇)(𝐹‘𝑥) ∈ (𝑌‘(𝐹 “ 𝑇)))
5		ssralv 4004	. . 3 ⊢ (𝑆 ⊆ (𝑍‘𝑇) → (∀𝑥 ∈ (𝑍‘𝑇)(𝐹‘𝑥) ∈ (𝑌‘(𝐹 “ 𝑇)) → ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) ∈ (𝑌‘(𝐹 “ 𝑇))))
6	4, 5	mpan9 506	. 2 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) ∈ (𝑌‘(𝐹 “ 𝑇)))
7		eqid 2729	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
8		eqid 2729	. . . . . 6 ⊢ (Base‘𝐻) = (Base‘𝐻)
9	7, 8	mhmf 18663	. . . . 5 ⊢ (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
10	9	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
11	10	ffund 6656	. . 3 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → Fun 𝐹)
12		simpr 484	. . . . 5 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → 𝑆 ⊆ (𝑍‘𝑇))
13	7, 1	cntzssv 19207	. . . . 5 ⊢ (𝑍‘𝑇) ⊆ (Base‘𝐺)
14	12, 13	sstrdi 3948	. . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → 𝑆 ⊆ (Base‘𝐺))
15	10	fdmd 6662	. . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → dom 𝐹 = (Base‘𝐺))
16	14, 15	sseqtrrd 3973	. . 3 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → 𝑆 ⊆ dom 𝐹)
17		funimass4 6887	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑆 ⊆ dom 𝐹) → ((𝐹 “ 𝑆) ⊆ (𝑌‘(𝐹 “ 𝑇)) ↔ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) ∈ (𝑌‘(𝐹 “ 𝑇))))
18	11, 16, 17	syl2anc 584	. 2 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → ((𝐹 “ 𝑆) ⊆ (𝑌‘(𝐹 “ 𝑇)) ↔ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥) ∈ (𝑌‘(𝐹 “ 𝑇))))
19	6, 18	mpbird 257	1 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → (𝐹 “ 𝑆) ⊆ (𝑌‘(𝐹 “ 𝑇)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3903  dom cdm 5619   “ cima 5622  Fun wfun 6476  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349  Basecbs 17120   MndHom cmhm 18655  Cntzccntz 19194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-map 8755  df-mhm 18657  df-cntz 19196

This theorem is referenced by:  gsumzmhm  19816  gsumzinv  19824