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Theorem resghm2b 19156
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 19140 . . 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 19140 . . 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 19072 . . . . . 6 (𝑋 ∈ (SubGrpβ€˜π‘‡) β†’ 𝑋 ∈ (SubMndβ€˜π‘‡))
6 resghm2.u . . . . . . 7 π‘ˆ = (𝑇 β†Ύs 𝑋)
76resmhm2b 18744 . . . . . 6 ((𝑋 ∈ (SubMndβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom π‘ˆ)))
85, 7sylan 579 . . . . 5 ((𝑋 ∈ (SubGrpβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom π‘ˆ)))
98adantl 481 . . . 4 ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrpβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋)) β†’ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom π‘ˆ)))
10 subgrcl 19055 . . . . . . 7 (𝑋 ∈ (SubGrpβ€˜π‘‡) β†’ 𝑇 ∈ Grp)
1110adantr 480 . . . . . 6 ((𝑋 ∈ (SubGrpβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ 𝑇 ∈ Grp)
12 ghmmhmb 19149 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) β†’ (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇))
1311, 12sylan2 592 . . . . 5 ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrpβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋)) β†’ (𝑆 GrpHom 𝑇) = (𝑆 MndHom 𝑇))
1413eleq2d 2813 . . . 4 ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrpβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋)) β†’ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 MndHom 𝑇)))
156subggrp 19053 . . . . . . 7 (𝑋 ∈ (SubGrpβ€˜π‘‡) β†’ π‘ˆ ∈ Grp)
1615adantr 480 . . . . . 6 ((𝑋 ∈ (SubGrpβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋) β†’ π‘ˆ ∈ Grp)
17 ghmmhmb 19149 . . . . . 6 ((𝑆 ∈ Grp ∧ π‘ˆ ∈ Grp) β†’ (𝑆 GrpHom π‘ˆ) = (𝑆 MndHom π‘ˆ))
1816, 17sylan2 592 . . . . 5 ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrpβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋)) β†’ (𝑆 GrpHom π‘ˆ) = (𝑆 MndHom π‘ˆ))
1918eleq2d 2813 . . . 4 ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrpβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋)) β†’ (𝐹 ∈ (𝑆 GrpHom π‘ˆ) ↔ 𝐹 ∈ (𝑆 MndHom π‘ˆ)))
209, 14, 193bitr4d 311 . . 3 ((𝑆 ∈ Grp ∧ (𝑋 ∈ (SubGrpβ€˜π‘‡) ∧ ran 𝐹 βŠ† 𝑋)) β†’ (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ (𝑆 GrpHom π‘ˆ)))
2120expcom 413 . 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 379 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 395   = wceq 1533   ∈ wcel 2098   βŠ† wss 3943  ran crn 5670  β€˜cfv 6536  (class class class)co 7404   β†Ύs cress 17179   MndHom cmhm 18708  SubMndcsubmnd 18709  Grpcgrp 18860  SubGrpcsubg 19044   GrpHom cghm 19135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  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 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-2 12276  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17151  df-ress 17180  df-plusg 17216  df-0g 17393  df-mgm 18570  df-sgrp 18649  df-mnd 18665  df-mhm 18710  df-submnd 18711  df-grp 18863  df-minusg 18864  df-subg 19047  df-ghm 19136
This theorem is referenced by:  ghmghmrn  19157  cayley  19331  pj1ghm2  19621  dpjghm2  19983  resrhm2b  20501  reslmhm2b  20899  m2cpmghm  22596
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