Theorem cntzmhm2 19329

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 19328	. . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑥 ∈ (𝑍‘𝑇)) → (𝐹‘𝑥) ∈ (𝑌‘(𝐹 “ 𝑇)))
4	3	ralrimiva 3133	. . 3 ⊢ (𝐹 ∈ (𝐺 MndHom 𝐻) → ∀𝑥 ∈ (𝑍‘𝑇)(𝐹‘𝑥) ∈ (𝑌‘(𝐹 “ 𝑇)))
5		ssralv 4032	. . 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 2734	. . . . . 6 ⊢ (Base‘𝐺) = (Base‘𝐺)
8		eqid 2734	. . . . . 6 ⊢ (Base‘𝐻) = (Base‘𝐻)
9	7, 8	mhmf 18771	. . . . 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 6720	. . 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 19315	. . . . 5 ⊢ (𝑍‘𝑇) ⊆ (Base‘𝐺)
14	12, 13	sstrdi 3976	. . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → 𝑆 ⊆ (Base‘𝐺))
15	10	fdmd 6726	. . . 4 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → dom 𝐹 = (Base‘𝐺))
16	14, 15	sseqtrrd 4001	. . 3 ⊢ ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍‘𝑇)) → 𝑆 ⊆ dom 𝐹)
17		funimass4 6953	. . 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 1539   ∈ wcel 2107  ∀wral 3050   ⊆ wss 3931  dom cdm 5665   “ cima 5668  Fun wfun 6535  ⟶wf 6537  ‘cfv 6541  (class class class)co 7413  Basecbs 17229   MndHom cmhm 18763  Cntzccntz 19302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7416  df-oprab 7417  df-mpo 7418  df-map 8850  df-mhm 18765  df-cntz 19304

This theorem is referenced by:  gsumzmhm  19923  gsumzinv  19931