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Theorem cntzmhm2 18465
 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 18464 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑥 ∈ (𝑍𝑇)) → (𝐹𝑥) ∈ (𝑌‘(𝐹𝑇)))
43ralrimiva 3152 . . 3 (𝐹 ∈ (𝐺 MndHom 𝐻) → ∀𝑥 ∈ (𝑍𝑇)(𝐹𝑥) ∈ (𝑌‘(𝐹𝑇)))
5 ssralv 3984 . . 3 (𝑆 ⊆ (𝑍𝑇) → (∀𝑥 ∈ (𝑍𝑇)(𝐹𝑥) ∈ (𝑌‘(𝐹𝑇)) → ∀𝑥𝑆 (𝐹𝑥) ∈ (𝑌‘(𝐹𝑇))))
64, 5mpan9 510 . 2 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → ∀𝑥𝑆 (𝐹𝑥) ∈ (𝑌‘(𝐹𝑇)))
7 eqid 2801 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
8 eqid 2801 . . . . . 6 (Base‘𝐻) = (Base‘𝐻)
97, 8mhmf 17956 . . . . 5 (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
109adantr 484 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
1110ffund 6495 . . 3 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → Fun 𝐹)
12 simpr 488 . . . . 5 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → 𝑆 ⊆ (𝑍𝑇))
137, 1cntzssv 18453 . . . . 5 (𝑍𝑇) ⊆ (Base‘𝐺)
1412, 13sstrdi 3930 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → 𝑆 ⊆ (Base‘𝐺))
1510fdmd 6501 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → dom 𝐹 = (Base‘𝐺))
1614, 15sseqtrrd 3959 . . 3 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → 𝑆 ⊆ dom 𝐹)
17 funimass4 6709 . . 3 ((Fun 𝐹𝑆 ⊆ dom 𝐹) → ((𝐹𝑆) ⊆ (𝑌‘(𝐹𝑇)) ↔ ∀𝑥𝑆 (𝐹𝑥) ∈ (𝑌‘(𝐹𝑇))))
1811, 16, 17syl2anc 587 . 2 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → ((𝐹𝑆) ⊆ (𝑌‘(𝐹𝑇)) ↔ ∀𝑥𝑆 (𝐹𝑥) ∈ (𝑌‘(𝐹𝑇))))
196, 18mpbird 260 1 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → (𝐹𝑆) ⊆ (𝑌‘(𝐹𝑇)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2112  ∀wral 3109   ⊆ wss 3884  dom cdm 5523   “ cima 5526  Fun wfun 6322  ⟶wf 6324  ‘cfv 6328  (class class class)co 7139  Basecbs 16478   MndHom cmhm 17949  Cntzccntz 18440 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-map 8395  df-mhm 17951  df-cntz 18442 This theorem is referenced by:  gsumzmhm  19053  gsumzinv  19061
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