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Theorem cntzmhm2 18408
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 ⊆ (𝑍𝑇)) → (𝐹𝑆) ⊆ (𝑌‘(𝐹𝑇)))

Proof of Theorem cntzmhm2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 cntzmhm.z . . . . 5 𝑍 = (Cntz‘𝐺)
2 cntzmhm.y . . . . 5 𝑌 = (Cntz‘𝐻)
31, 2cntzmhm 18407 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑥 ∈ (𝑍𝑇)) → (𝐹𝑥) ∈ (𝑌‘(𝐹𝑇)))
43ralrimiva 3179 . . 3 (𝐹 ∈ (𝐺 MndHom 𝐻) → ∀𝑥 ∈ (𝑍𝑇)(𝐹𝑥) ∈ (𝑌‘(𝐹𝑇)))
5 ssralv 4030 . . 3 (𝑆 ⊆ (𝑍𝑇) → (∀𝑥 ∈ (𝑍𝑇)(𝐹𝑥) ∈ (𝑌‘(𝐹𝑇)) → ∀𝑥𝑆 (𝐹𝑥) ∈ (𝑌‘(𝐹𝑇))))
64, 5mpan9 507 . 2 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → ∀𝑥𝑆 (𝐹𝑥) ∈ (𝑌‘(𝐹𝑇)))
7 eqid 2818 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
8 eqid 2818 . . . . . 6 (Base‘𝐻) = (Base‘𝐻)
97, 8mhmf 17949 . . . . 5 (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
109adantr 481 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
1110ffund 6511 . . 3 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → Fun 𝐹)
12 simpr 485 . . . . 5 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → 𝑆 ⊆ (𝑍𝑇))
137, 1cntzssv 18396 . . . . 5 (𝑍𝑇) ⊆ (Base‘𝐺)
1412, 13sstrdi 3976 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → 𝑆 ⊆ (Base‘𝐺))
1510fdmd 6516 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → dom 𝐹 = (Base‘𝐺))
1614, 15sseqtrrd 4005 . . 3 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → 𝑆 ⊆ dom 𝐹)
17 funimass4 6723 . . 3 ((Fun 𝐹𝑆 ⊆ dom 𝐹) → ((𝐹𝑆) ⊆ (𝑌‘(𝐹𝑇)) ↔ ∀𝑥𝑆 (𝐹𝑥) ∈ (𝑌‘(𝐹𝑇))))
1811, 16, 17syl2anc 584 . 2 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → ((𝐹𝑆) ⊆ (𝑌‘(𝐹𝑇)) ↔ ∀𝑥𝑆 (𝐹𝑥) ∈ (𝑌‘(𝐹𝑇))))
196, 18mpbird 258 1 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → (𝐹𝑆) ⊆ (𝑌‘(𝐹𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wss 3933  dom cdm 5548  cima 5551  Fun wfun 6342  wf 6344  cfv 6348  (class class class)co 7145  Basecbs 16471   MndHom cmhm 17942  Cntzccntz 18383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-mhm 17944  df-cntz 18385
This theorem is referenced by:  gsumzmhm  18986  gsumzinv  18994
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