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Theorem cntzmhm2 17978
 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 17977 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑥 ∈ (𝑍𝑇)) → (𝐹𝑥) ∈ (𝑌‘(𝐹𝑇)))
43ralrimiva 3114 . . 3 (𝐹 ∈ (𝐺 MndHom 𝐻) → ∀𝑥 ∈ (𝑍𝑇)(𝐹𝑥) ∈ (𝑌‘(𝐹𝑇)))
5 ssralv 3813 . . 3 (𝑆 ⊆ (𝑍𝑇) → (∀𝑥 ∈ (𝑍𝑇)(𝐹𝑥) ∈ (𝑌‘(𝐹𝑇)) → ∀𝑥𝑆 (𝐹𝑥) ∈ (𝑌‘(𝐹𝑇))))
64, 5mpan9 490 . 2 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → ∀𝑥𝑆 (𝐹𝑥) ∈ (𝑌‘(𝐹𝑇)))
7 eqid 2770 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
8 eqid 2770 . . . . . 6 (Base‘𝐻) = (Base‘𝐻)
97, 8mhmf 17547 . . . . 5 (𝐹 ∈ (𝐺 MndHom 𝐻) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
109adantr 466 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → 𝐹:(Base‘𝐺)⟶(Base‘𝐻))
11 ffun 6188 . . . 4 (𝐹:(Base‘𝐺)⟶(Base‘𝐻) → Fun 𝐹)
1210, 11syl 17 . . 3 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → Fun 𝐹)
13 simpr 471 . . . . 5 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → 𝑆 ⊆ (𝑍𝑇))
147, 1cntzssv 17967 . . . . 5 (𝑍𝑇) ⊆ (Base‘𝐺)
1513, 14syl6ss 3762 . . . 4 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → 𝑆 ⊆ (Base‘𝐺))
16 fdm 6191 . . . . 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 3789 . . 3 ((𝐹 ∈ (𝐺 MndHom 𝐻) ∧ 𝑆 ⊆ (𝑍𝑇)) → 𝑆 ⊆ dom 𝐹)
19 funimass4 6389 . . 3 ((Fun 𝐹𝑆 ⊆ dom 𝐹) → ((𝐹𝑆) ⊆ (𝑌‘(𝐹𝑇)) ↔ ∀𝑥𝑆 (𝐹𝑥) ∈ (𝑌‘(𝐹𝑇))))
2012, 18, 19syl2anc 565 . 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 382   = wceq 1630   ∈ wcel 2144  ∀wral 3060   ⊆ wss 3721  dom cdm 5249   “ cima 5252  Fun wfun 6025  ⟶wf 6027  ‘cfv 6031  (class class class)co 6792  Basecbs 16063   MndHom cmhm 17540  Cntzccntz 17954 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-map 8010  df-mhm 17542  df-cntz 17956 This theorem is referenced by:  gsumzmhm  18543  gsumzinv  18551
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