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Theorem resghm2b 19109
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 π‘ˆ = (𝑇 β†Ύs 𝑋)
Assertion
Ref Expression
resghm2b ((𝑋 ∈ (SubGrpβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom π‘ˆ)))
Proof of Theorem resghm2b
StepHypRef Expression
1 ghmgrp1 19093 . . 3 (𝐹 ∈ (𝑆 GrpHom 𝑇) β†’ 𝑆 ∈ Grp)
21a1i 11 . 2 ((𝑋 ∈ (SubGrpβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ (𝐹 ∈ (𝑆 GrpHom 𝑇) β†’ 𝑆 ∈ Grp))
3 ghmgrp1 19093 . . 3 (𝐹 ∈ (𝑆 GrpHom π‘ˆ) β†’ 𝑆 ∈ Grp)
43a1i 11 . 2 ((𝑋 ∈ (SubGrpβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ (𝐹 ∈ (𝑆 GrpHom π‘ˆ) β†’ 𝑆 ∈ Grp))
5 subgsubm 19027 . . . . . 6 (𝑋 ∈ (SubGrpβ€˜π‘‡) β†’ 𝑋 ∈ (SubMndβ€˜π‘‡))
6 resghm2.u . . . . . . 7 π‘ˆ = (𝑇 β†Ύs 𝑋)
76resmhm2b 18702 . . . . . 6 ((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom π‘ˆ)))
85, 7sylan 580 . . . . 5 ((𝑋 ∈ (SubGrpβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom π‘ˆ)))
98adantl 482 . . . 4 ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrpβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋)) β†’ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom π‘ˆ)))
10 subgrcl 19010 . . . . . . 7 (𝑋 ∈ (SubGrpβ€˜π‘‡) β†’ 𝑇 ∈ Grp)
1110adantr 481 . . . . . 6 ((𝑋 ∈ (SubGrpβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ 𝑇 ∈ Grp)
12 ghmmhmb 19102 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) β†’ (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇))
1311, 12sylan2 593 . . . . 5 ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrpβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋)) β†’ (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇))
1413eleq2d 2819 . . . 4 ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrpβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋)) β†’ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑇)))
156subggrp 19008 . . . . . . 7 (𝑋 ∈ (SubGrpβ€˜π‘‡) β†’ π‘ˆ ∈ Grp)
1615adantr 481 . . . . . 6 ((𝑋 ∈ (SubGrpβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ π‘ˆ ∈ Grp)
17 ghmmhmb 19102 . . . . . 6 ((𝑆 ∈ Grp ∧ π‘ˆ ∈ Grp) β†’ (𝑆 GrpHom π‘ˆ) = (𝑆 MndHom π‘ˆ))
1816, 17sylan2 593 . . . . 5 ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrpβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋)) β†’ (𝑆 GrpHom π‘ˆ) = (𝑆 MndHom π‘ˆ))
1918eleq2d 2819 . . . 4 ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrpβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋)) β†’ (𝐹 ∈ (𝑆 GrpHom π‘ˆ) ↔ 𝐹 ∈ (𝑆 MndHom π‘ˆ)))
209, 14, 193bitr4d 310 . . 3 ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrpβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋)) β†’ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom π‘ˆ)))
2120expcom 414 . 2 ((𝑋 ∈ (SubGrpβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ (𝑆 ∈ Grp β†’ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom π‘ˆ))))
222, 4, 21pm5.21ndd 380 1 ((𝑋 ∈ (SubGrpβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom π‘ˆ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   βŠ† wss 3948  ran crn 5677  β€˜cfv 6543  (class class class)co 7408   β†Ύs cress 17172   MndHom cmhm 18668  SubMndcsubmnd 18669  Grpcgrp 18818  SubGrpcsubg 18999   GrpHom cghm 19088
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-0g 17386  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-mhm 18670  df-submnd 18671  df-grp 18821  df-minusg 18822  df-subg 19002  df-ghm 19089
This theorem is referenced by:  ghmghmrn  19110  cayley  19281  pj1ghm2  19571  dpjghm2  19933  resrhm2b  20348  reslmhm2b  20664  m2cpmghm  22245
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