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Theorem funcringcsetclem8ALTV 46918
Description: Lemma 8 for funcringcsetcALTV 46920. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
funcringcsetcALTV.r 𝑅 = (RingCatALTV‘𝑈)
funcringcsetcALTV.s 𝑆 = (SetCat‘𝑈)
funcringcsetcALTV.b 𝐵 = (Base‘𝑅)
funcringcsetcALTV.c 𝐶 = (Base‘𝑆)
funcringcsetcALTV.u (𝜑𝑈 ∈ WUni)
funcringcsetcALTV.f (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
funcringcsetcALTV.g (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
Assertion
Ref Expression
funcringcsetclem8ALTV ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝑅)𝑌)⟶((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)))
Distinct variable groups:   𝑥,𝐵   𝑥,𝑋   𝜑,𝑥   𝑥,𝐶   𝑦,𝐵,𝑥   𝑦,𝑋   𝑥,𝑌,𝑦   𝜑,𝑦
Allowed substitution hints:   𝐶(𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem funcringcsetclem8ALTV
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 f1oi 6869 . . . 4 ( I ↾ (𝑋 RingHom 𝑌)):(𝑋 RingHom 𝑌)–1-1-onto→(𝑋 RingHom 𝑌)
2 f1of 6831 . . . 4 (( I ↾ (𝑋 RingHom 𝑌)):(𝑋 RingHom 𝑌)–1-1-onto→(𝑋 RingHom 𝑌) → ( I ↾ (𝑋 RingHom 𝑌)):(𝑋 RingHom 𝑌)⟶(𝑋 RingHom 𝑌))
31, 2mp1i 13 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ( I ↾ (𝑋 RingHom 𝑌)):(𝑋 RingHom 𝑌)⟶(𝑋 RingHom 𝑌))
4 eqid 2733 . . . . . 6 (Base‘𝑋) = (Base‘𝑋)
5 eqid 2733 . . . . . 6 (Base‘𝑌) = (Base‘𝑌)
64, 5rhmf 20256 . . . . 5 (𝑓 ∈ (𝑋 RingHom 𝑌) → 𝑓:(Base‘𝑋)⟶(Base‘𝑌))
7 fvex 6902 . . . . . . . . . 10 (Base‘𝑌) ∈ V
8 fvex 6902 . . . . . . . . . 10 (Base‘𝑋) ∈ V
97, 8pm3.2i 472 . . . . . . . . 9 ((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V)
10 elmapg 8830 . . . . . . . . . 10 (((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V) → (𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ↔ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)))
1110bicomd 222 . . . . . . . . 9 (((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V) → (𝑓:(Base‘𝑋)⟶(Base‘𝑌) ↔ 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))))
129, 11mp1i 13 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑓:(Base‘𝑋)⟶(Base‘𝑌) ↔ 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))))
1312biimpa 478 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)))
14 simpr 486 . . . . . . . . . 10 ((𝑋𝐵𝑌𝐵) → 𝑌𝐵)
15 funcringcsetcALTV.r . . . . . . . . . . 11 𝑅 = (RingCatALTV‘𝑈)
16 funcringcsetcALTV.s . . . . . . . . . . 11 𝑆 = (SetCat‘𝑈)
17 funcringcsetcALTV.b . . . . . . . . . . 11 𝐵 = (Base‘𝑅)
18 funcringcsetcALTV.c . . . . . . . . . . 11 𝐶 = (Base‘𝑆)
19 funcringcsetcALTV.u . . . . . . . . . . 11 (𝜑𝑈 ∈ WUni)
20 funcringcsetcALTV.f . . . . . . . . . . 11 (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
2115, 16, 17, 18, 19, 20funcringcsetclem1ALTV 46911 . . . . . . . . . 10 ((𝜑𝑌𝐵) → (𝐹𝑌) = (Base‘𝑌))
2214, 21sylan2 594 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑌) = (Base‘𝑌))
23 simpl 484 . . . . . . . . . 10 ((𝑋𝐵𝑌𝐵) → 𝑋𝐵)
2415, 16, 17, 18, 19, 20funcringcsetclem1ALTV 46911 . . . . . . . . . 10 ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
2523, 24sylan2 594 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑋) = (Base‘𝑋))
2622, 25oveq12d 7424 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((𝐹𝑌) ↑m (𝐹𝑋)) = ((Base‘𝑌) ↑m (Base‘𝑋)))
2726adantr 482 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → ((𝐹𝑌) ↑m (𝐹𝑋)) = ((Base‘𝑌) ↑m (Base‘𝑋)))
2813, 27eleqtrrd 2837 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → 𝑓 ∈ ((𝐹𝑌) ↑m (𝐹𝑋)))
2928ex 414 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑓:(Base‘𝑋)⟶(Base‘𝑌) → 𝑓 ∈ ((𝐹𝑌) ↑m (𝐹𝑋))))
306, 29syl5 34 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑓 ∈ (𝑋 RingHom 𝑌) → 𝑓 ∈ ((𝐹𝑌) ↑m (𝐹𝑋))))
3130ssrdv 3988 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 RingHom 𝑌) ⊆ ((𝐹𝑌) ↑m (𝐹𝑋)))
323, 31fssd 6733 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ( I ↾ (𝑋 RingHom 𝑌)):(𝑋 RingHom 𝑌)⟶((𝐹𝑌) ↑m (𝐹𝑋)))
33 funcringcsetcALTV.g . . . 4 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
3415, 16, 17, 18, 19, 20, 33funcringcsetclem5ALTV 46915 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑋 RingHom 𝑌)))
3519adantr 482 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑈 ∈ WUni)
36 eqid 2733 . . . 4 (Hom ‘𝑅) = (Hom ‘𝑅)
3723adantl 483 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
3814adantl 483 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
3915, 17, 35, 36, 37, 38ringchomALTV 46900 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(Hom ‘𝑅)𝑌) = (𝑋 RingHom 𝑌))
40 eqid 2733 . . . 4 (Hom ‘𝑆) = (Hom ‘𝑆)
4115, 16, 17, 18, 19, 20funcringcsetclem2ALTV 46912 . . . . 5 ((𝜑𝑋𝐵) → (𝐹𝑋) ∈ 𝑈)
4223, 41sylan2 594 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑋) ∈ 𝑈)
4315, 16, 17, 18, 19, 20funcringcsetclem2ALTV 46912 . . . . 5 ((𝜑𝑌𝐵) → (𝐹𝑌) ∈ 𝑈)
4414, 43sylan2 594 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑌) ∈ 𝑈)
4516, 35, 40, 42, 44setchom 18027 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)) = ((𝐹𝑌) ↑m (𝐹𝑋)))
4634, 39, 45feq123d 6704 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((𝑋𝐺𝑌):(𝑋(Hom ‘𝑅)𝑌)⟶((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)) ↔ ( I ↾ (𝑋 RingHom 𝑌)):(𝑋 RingHom 𝑌)⟶((𝐹𝑌) ↑m (𝐹𝑋))))
4732, 46mpbird 257 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝑅)𝑌)⟶((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  cmpt 5231   I cid 5573  cres 5678  wf 6537  1-1-ontowf1o 6540  cfv 6541  (class class class)co 7406  cmpo 7408  m cmap 8817  WUnicwun 10692  Basecbs 17141  Hom chom 17205  SetCatcsetc 18022   RingHom crh 20241  RingCatALTVcringcALTV 46856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  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-tp 4633  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 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-er 8700  df-map 8819  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-wun 10694  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-7 12277  df-8 12278  df-9 12279  df-n0 12470  df-z 12556  df-dec 12675  df-uz 12820  df-fz 13482  df-struct 17077  df-sets 17094  df-slot 17112  df-ndx 17124  df-base 17142  df-plusg 17207  df-hom 17218  df-cco 17219  df-0g 17384  df-setc 18023  df-mhm 18668  df-ghm 19085  df-mgp 19983  df-ur 20000  df-ring 20052  df-rnghom 20244  df-ringcALTV 46858
This theorem is referenced by:  funcringcsetcALTV  46920
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