MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cntzmhm2 Structured version   Visualization version   GIF version

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
  Copyright terms: Public domain W3C validator