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Theorem resghm2b 17594
Description: Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 13-Jan-2015.) (Revised by Mario Carneiro, 18-Jun-2015.)
Hypothesis
Ref Expression
resghm2.u 𝑈 = (𝑇s 𝑋)
Assertion
Ref Expression
resghm2b ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))

Proof of Theorem resghm2b
StepHypRef Expression
1 ghmgrp1 17578 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp)
21a1i 11 . 2 ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝑆 ∈ Grp))
3 ghmgrp1 17578 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑈) → 𝑆 ∈ Grp)
43a1i 11 . 2 ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑈) → 𝑆 ∈ Grp))
5 subgsubm 17532 . . . . . 6 (𝑋 ∈ (SubGrp‘𝑇) → 𝑋 ∈ (SubMnd‘𝑇))
6 resghm2.u . . . . . . 7 𝑈 = (𝑇s 𝑋)
76resmhm2b 17277 . . . . . 6 ((𝑋 ∈ (SubMnd‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈)))
85, 7sylan 488 . . . . 5 ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈)))
98adantl 482 . . . 4 ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹𝑋)) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈)))
10 subgrcl 17515 . . . . . . 7 (𝑋 ∈ (SubGrp‘𝑇) → 𝑇 ∈ Grp)
1110adantr 481 . . . . . 6 ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹𝑋) → 𝑇 ∈ Grp)
12 ghmmhmb 17587 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇))
1311, 12sylan2 491 . . . . 5 ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹𝑋)) → (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇))
1413eleq2d 2689 . . . 4 ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹𝑋)) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑇)))
156subggrp 17513 . . . . . . 7 (𝑋 ∈ (SubGrp‘𝑇) → 𝑈 ∈ Grp)
1615adantr 481 . . . . . 6 ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹𝑋) → 𝑈 ∈ Grp)
17 ghmmhmb 17587 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑈 ∈ Grp) → (𝑆 GrpHom 𝑈) = (𝑆 MndHom 𝑈))
1816, 17sylan2 491 . . . . 5 ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹𝑋)) → (𝑆 GrpHom 𝑈) = (𝑆 MndHom 𝑈))
1918eleq2d 2689 . . . 4 ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹𝑋)) → (𝐹 ∈ (𝑆 GrpHom 𝑈) ↔ 𝐹 ∈ (𝑆 MndHom 𝑈)))
209, 14, 193bitr4d 300 . . 3 ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹𝑋)) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))
2120expcom 451 . 2 ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹𝑋) → (𝑆 ∈ Grp → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈))))
222, 4, 21pm5.21ndd 369 1 ((𝑋 ∈ (SubGrp‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom 𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wss 3560  ran crn 5080  cfv 5850  (class class class)co 6605  s cress 15777   MndHom cmhm 17249  SubMndcsubmnd 17250  Grpcgrp 17338  SubGrpcsubg 17504   GrpHom cghm 17573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-map 7805  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-ress 15783  df-plusg 15870  df-0g 16018  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-mhm 17251  df-submnd 17252  df-grp 17341  df-minusg 17342  df-subg 17507  df-ghm 17574
This theorem is referenced by:  ghmghmrn  17595  cayley  17750  pj1ghm2  18033  dpjghm2  18379  reslmhm2b  18968  m2cpmghm  20463
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