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Theorem cntzmhm2 17766
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 17765 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑥 ∈ (𝑍𝑇)) → (𝐹𝑥) ∈ (𝑌‘(𝐹𝑇)))
43ralrimiva 2965 . . 3 (𝐹 ∈ (𝐺 MndHom 𝐻) → ∀𝑥 ∈ (𝑍𝑇)(𝐹𝑥) ∈ (𝑌‘(𝐹𝑇)))
5 ssralv 3664 . . 3 (𝑆 ⊆ (𝑍𝑇) → (∀𝑥 ∈ (𝑍𝑇)(𝐹𝑥) ∈ (𝑌‘(𝐹𝑇)) → ∀𝑥𝑆 (𝐹𝑥) ∈ (𝑌‘(𝐹𝑇))))
64, 5mpan9 486 . 2 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → ∀𝑥𝑆 (𝐹𝑥) ∈ (𝑌‘(𝐹𝑇)))
7 eqid 2621 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
8 eqid 2621 . . . . . 6 (Base‘𝐻) = (Base‘𝐻)
97, 8mhmf 17334 . . . . 5 (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
109adantr 481 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
11 ffun 6046 . . . 4 (𝐹:(Base‘𝐺)⟶(Base‘𝐻) → Fun 𝐹)
1210, 11syl 17 . . 3 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → Fun 𝐹)
13 simpr 477 . . . . 5 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → 𝑆 ⊆ (𝑍𝑇))
147, 1cntzssv 17755 . . . . 5 (𝑍𝑇) ⊆ (Base‘𝐺)
1513, 14syl6ss 3613 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → 𝑆 ⊆ (Base‘𝐺))
16 fdm 6049 . . . . 5 (𝐹:(Base‘𝐺)⟶(Base‘𝐻) → dom 𝐹 = (Base‘𝐺))
1710, 16syl 17 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → dom 𝐹 = (Base‘𝐺))
1815, 17sseqtr4d 3640 . . 3 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → 𝑆 ⊆ dom 𝐹)
19 funimass4 6245 . . 3 ((Fun 𝐹𝑆 ⊆ dom 𝐹) → ((𝐹𝑆) ⊆ (𝑌‘(𝐹𝑇)) ↔ ∀𝑥𝑆 (𝐹𝑥) ∈ (𝑌‘(𝐹𝑇))))
2012, 18, 19syl2anc 693 . 2 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → ((𝐹𝑆) ⊆ (𝑌‘(𝐹𝑇)) ↔ ∀𝑥𝑆 (𝐹𝑥) ∈ (𝑌‘(𝐹𝑇))))
216, 20mpbird 247 1 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → (𝐹𝑆) ⊆ (𝑌‘(𝐹𝑇)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  wral 2911  wss 3572  dom cdm 5112  cima 5115  Fun wfun 5880  wf 5882  cfv 5886  (class class class)co 6647  Basecbs 15851   MndHom cmhm 17327  Cntzccntz 17742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-map 7856  df-mhm 17329  df-cntz 17744
This theorem is referenced by:  gsumzmhm  18331  gsumzinv  18339
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