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Theorem funcringcsetclem8ALTV 44169
Description: Lemma 8 for funcringcsetcALTV 44171. (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 6649 . . . 4 ( I ↾ (𝑋 RingHom 𝑌)):(𝑋 RingHom 𝑌)–1-1-onto→(𝑋 RingHom 𝑌)
2 f1of 6612 . . . 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 2826 . . . . . 6 (Base‘𝑋) = (Base‘𝑋)
5 eqid 2826 . . . . . 6 (Base‘𝑌) = (Base‘𝑌)
64, 5rhmf 19398 . . . . 5 (𝑓 ∈ (𝑋 RingHom 𝑌) → 𝑓:(Base‘𝑋)⟶(Base‘𝑌))
7 fvex 6680 . . . . . . . . . 10 (Base‘𝑌) ∈ V
8 fvex 6680 . . . . . . . . . 10 (Base‘𝑋) ∈ V
97, 8pm3.2i 471 . . . . . . . . 9 ((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V)
10 elmapg 8409 . . . . . . . . . 10 (((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V) → (𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ↔ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)))
1110bicomd 224 . . . . . . . . 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 477 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)))
14 simpr 485 . . . . . . . . . 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 44162 . . . . . . . . . 10 ((𝜑𝑌𝐵) → (𝐹𝑌) = (Base‘𝑌))
2214, 21sylan2 592 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑌) = (Base‘𝑌))
23 simpl 483 . . . . . . . . . 10 ((𝑋𝐵𝑌𝐵) → 𝑋𝐵)
2415, 16, 17, 18, 19, 20funcringcsetclem1ALTV 44162 . . . . . . . . . 10 ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
2523, 24sylan2 592 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑋) = (Base‘𝑋))
2622, 25oveq12d 7166 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((𝐹𝑌) ↑m (𝐹𝑋)) = ((Base‘𝑌) ↑m (Base‘𝑋)))
2726adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → ((𝐹𝑌) ↑m (𝐹𝑋)) = ((Base‘𝑌) ↑m (Base‘𝑋)))
2813, 27eleqtrrd 2921 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → 𝑓 ∈ ((𝐹𝑌) ↑m (𝐹𝑋)))
2928ex 413 . . . . 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 3977 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 RingHom 𝑌) ⊆ ((𝐹𝑌) ↑m (𝐹𝑋)))
323, 31fssd 6525 . 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 44166 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑋 RingHom 𝑌)))
3519adantr 481 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑈 ∈ WUni)
36 eqid 2826 . . . 4 (Hom ‘𝑅) = (Hom ‘𝑅)
3723adantl 482 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
3814adantl 482 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
3915, 17, 35, 36, 37, 38ringchomALTV 44151 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(Hom ‘𝑅)𝑌) = (𝑋 RingHom 𝑌))
40 eqid 2826 . . . 4 (Hom ‘𝑆) = (Hom ‘𝑆)
4115, 16, 17, 18, 19, 20funcringcsetclem2ALTV 44163 . . . . 5 ((𝜑𝑋𝐵) → (𝐹𝑋) ∈ 𝑈)
4223, 41sylan2 592 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑋) ∈ 𝑈)
4315, 16, 17, 18, 19, 20funcringcsetclem2ALTV 44163 . . . . 5 ((𝜑𝑌𝐵) → (𝐹𝑌) ∈ 𝑈)
4414, 43sylan2 592 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑌) ∈ 𝑈)
4516, 35, 40, 42, 44setchom 17330 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)) = ((𝐹𝑌) ↑m (𝐹𝑋)))
4634, 39, 45feq123d 6500 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((𝑋𝐺𝑌):(𝑋(Hom ‘𝑅)𝑌)⟶((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)) ↔ ( I ↾ (𝑋 RingHom 𝑌)):(𝑋 RingHom 𝑌)⟶((𝐹𝑌) ↑m (𝐹𝑋))))
4732, 46mpbird 258 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝑅)𝑌)⟶((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  Vcvv 3500  cmpt 5143   I cid 5458  cres 5556  wf 6348  1-1-ontowf1o 6351  cfv 6352  (class class class)co 7148  cmpo 7150  m cmap 8396  WUnicwun 10111  Basecbs 16473  Hom chom 16566  SetCatcsetc 17325   RingHom crh 19384  RingCatALTVcringcALTV 44107
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-map 8398  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-wun 10113  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-3 11690  df-4 11691  df-5 11692  df-6 11693  df-7 11694  df-8 11695  df-9 11696  df-n0 11887  df-z 11971  df-dec 12088  df-uz 12233  df-fz 12883  df-struct 16475  df-ndx 16476  df-slot 16477  df-base 16479  df-sets 16480  df-plusg 16568  df-hom 16579  df-cco 16580  df-0g 16705  df-setc 17326  df-mhm 17944  df-ghm 18286  df-mgp 19160  df-ur 19172  df-ring 19219  df-rnghom 19387  df-ringcALTV 44109
This theorem is referenced by:  funcringcsetcALTV  44171
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