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Theorem funcringcsetclem8ALTV 46917
Description: Lemma 8 for funcringcsetcALTV 46919. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
funcringcsetcALTV.r 𝑅 = (RingCatALTVβ€˜π‘ˆ)
funcringcsetcALTV.s 𝑆 = (SetCatβ€˜π‘ˆ)
funcringcsetcALTV.b 𝐡 = (Baseβ€˜π‘…)
funcringcsetcALTV.c 𝐢 = (Baseβ€˜π‘†)
funcringcsetcALTV.u (πœ‘ β†’ π‘ˆ ∈ WUni)
funcringcsetcALTV.f (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)))
funcringcsetcALTV.g (πœ‘ β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RingHom 𝑦))))
Assertion
Ref Expression
funcringcsetclem8ALTV ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (π‘‹πΊπ‘Œ):(𝑋(Hom β€˜π‘…)π‘Œ)⟢((πΉβ€˜π‘‹)(Hom β€˜π‘†)(πΉβ€˜π‘Œ)))
Distinct variable groups:   π‘₯,𝐡   π‘₯,𝑋   πœ‘,π‘₯   π‘₯,𝐢   𝑦,𝐡,π‘₯   𝑦,𝑋   π‘₯,π‘Œ,𝑦   πœ‘,𝑦
Allowed substitution hints:   𝐢(𝑦)   𝑅(π‘₯,𝑦)   𝑆(π‘₯,𝑦)   π‘ˆ(π‘₯,𝑦)   𝐹(π‘₯,𝑦)   𝐺(π‘₯,𝑦)
Proof of Theorem funcringcsetclem8ALTV
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 f1oi 6868 . . . 4 ( I β†Ύ (𝑋 RingHom π‘Œ)):(𝑋 RingHom π‘Œ)–1-1-ontoβ†’(𝑋 RingHom π‘Œ)
2 f1of 6830 . . . 4 (( I β†Ύ (𝑋 RingHom π‘Œ)):(𝑋 RingHom π‘Œ)–1-1-ontoβ†’(𝑋 RingHom π‘Œ) β†’ ( I β†Ύ (𝑋 RingHom π‘Œ)):(𝑋 RingHom π‘Œ)⟢(𝑋 RingHom π‘Œ))
31, 2mp1i 13 . . 3 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ ( I β†Ύ (𝑋 RingHom π‘Œ)):(𝑋 RingHom π‘Œ)⟢(𝑋 RingHom π‘Œ))
4 eqid 2732 . . . . . 6 (Baseβ€˜π‘‹) = (Baseβ€˜π‘‹)
5 eqid 2732 . . . . . 6 (Baseβ€˜π‘Œ) = (Baseβ€˜π‘Œ)
64, 5rhmf 20255 . . . . 5 (𝑓 ∈ (𝑋 RingHom π‘Œ) β†’ 𝑓:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ))
7 fvex 6901 . . . . . . . . . 10 (Baseβ€˜π‘Œ) ∈ V
8 fvex 6901 . . . . . . . . . 10 (Baseβ€˜π‘‹) ∈ V
97, 8pm3.2i 471 . . . . . . . . 9 ((Baseβ€˜π‘Œ) ∈ V ∧ (Baseβ€˜π‘‹) ∈ V)
10 elmapg 8829 . . . . . . . . . 10 (((Baseβ€˜π‘Œ) ∈ V ∧ (Baseβ€˜π‘‹) ∈ V) β†’ (𝑓 ∈ ((Baseβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)) ↔ 𝑓:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ)))
1110bicomd 222 . . . . . . . . 9 (((Baseβ€˜π‘Œ) ∈ V ∧ (Baseβ€˜π‘‹) ∈ V) β†’ (𝑓:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ) ↔ 𝑓 ∈ ((Baseβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹))))
129, 11mp1i 13 . . . . . . . 8 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (𝑓:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ) ↔ 𝑓 ∈ ((Baseβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹))))
1312biimpa 477 . . . . . . 7 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) ∧ 𝑓:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ)) β†’ 𝑓 ∈ ((Baseβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)))
14 simpr 485 . . . . . . . . . 10 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ π‘Œ ∈ 𝐡)
15 funcringcsetcALTV.r . . . . . . . . . . 11 𝑅 = (RingCatALTVβ€˜π‘ˆ)
16 funcringcsetcALTV.s . . . . . . . . . . 11 𝑆 = (SetCatβ€˜π‘ˆ)
17 funcringcsetcALTV.b . . . . . . . . . . 11 𝐡 = (Baseβ€˜π‘…)
18 funcringcsetcALTV.c . . . . . . . . . . 11 𝐢 = (Baseβ€˜π‘†)
19 funcringcsetcALTV.u . . . . . . . . . . 11 (πœ‘ β†’ π‘ˆ ∈ WUni)
20 funcringcsetcALTV.f . . . . . . . . . . 11 (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)))
2115, 16, 17, 18, 19, 20funcringcsetclem1ALTV 46910 . . . . . . . . . 10 ((πœ‘ ∧ π‘Œ ∈ 𝐡) β†’ (πΉβ€˜π‘Œ) = (Baseβ€˜π‘Œ))
2214, 21sylan2 593 . . . . . . . . 9 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (πΉβ€˜π‘Œ) = (Baseβ€˜π‘Œ))
23 simpl 483 . . . . . . . . . 10 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ 𝑋 ∈ 𝐡)
2415, 16, 17, 18, 19, 20funcringcsetclem1ALTV 46910 . . . . . . . . . 10 ((πœ‘ ∧ 𝑋 ∈ 𝐡) β†’ (πΉβ€˜π‘‹) = (Baseβ€˜π‘‹))
2523, 24sylan2 593 . . . . . . . . 9 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (πΉβ€˜π‘‹) = (Baseβ€˜π‘‹))
2622, 25oveq12d 7423 . . . . . . . 8 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ ((πΉβ€˜π‘Œ) ↑m (πΉβ€˜π‘‹)) = ((Baseβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)))
2726adantr 481 . . . . . . 7 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) ∧ 𝑓:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ)) β†’ ((πΉβ€˜π‘Œ) ↑m (πΉβ€˜π‘‹)) = ((Baseβ€˜π‘Œ) ↑m (Baseβ€˜π‘‹)))
2813, 27eleqtrrd 2836 . . . . . 6 (((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) ∧ 𝑓:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ)) β†’ 𝑓 ∈ ((πΉβ€˜π‘Œ) ↑m (πΉβ€˜π‘‹)))
2928ex 413 . . . . 5 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (𝑓:(Baseβ€˜π‘‹)⟢(Baseβ€˜π‘Œ) β†’ 𝑓 ∈ ((πΉβ€˜π‘Œ) ↑m (πΉβ€˜π‘‹))))
306, 29syl5 34 . . . 4 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (𝑓 ∈ (𝑋 RingHom π‘Œ) β†’ 𝑓 ∈ ((πΉβ€˜π‘Œ) ↑m (πΉβ€˜π‘‹))))
3130ssrdv 3987 . . 3 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (𝑋 RingHom π‘Œ) βŠ† ((πΉβ€˜π‘Œ) ↑m (πΉβ€˜π‘‹)))
323, 31fssd 6732 . 2 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ ( I β†Ύ (𝑋 RingHom π‘Œ)):(𝑋 RingHom π‘Œ)⟢((πΉβ€˜π‘Œ) ↑m (πΉβ€˜π‘‹)))
33 funcringcsetcALTV.g . . . 4 (πœ‘ β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ (π‘₯ RingHom 𝑦))))
3415, 16, 17, 18, 19, 20, 33funcringcsetclem5ALTV 46914 . . 3 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (π‘‹πΊπ‘Œ) = ( I β†Ύ (𝑋 RingHom π‘Œ)))
3519adantr 481 . . . 4 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ π‘ˆ ∈ WUni)
36 eqid 2732 . . . 4 (Hom β€˜π‘…) = (Hom β€˜π‘…)
3723adantl 482 . . . 4 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ 𝑋 ∈ 𝐡)
3814adantl 482 . . . 4 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ π‘Œ ∈ 𝐡)
3915, 17, 35, 36, 37, 38ringchomALTV 46899 . . 3 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (𝑋(Hom β€˜π‘…)π‘Œ) = (𝑋 RingHom π‘Œ))
40 eqid 2732 . . . 4 (Hom β€˜π‘†) = (Hom β€˜π‘†)
4115, 16, 17, 18, 19, 20funcringcsetclem2ALTV 46911 . . . . 5 ((πœ‘ ∧ 𝑋 ∈ 𝐡) β†’ (πΉβ€˜π‘‹) ∈ π‘ˆ)
4223, 41sylan2 593 . . . 4 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (πΉβ€˜π‘‹) ∈ π‘ˆ)
4315, 16, 17, 18, 19, 20funcringcsetclem2ALTV 46911 . . . . 5 ((πœ‘ ∧ π‘Œ ∈ 𝐡) β†’ (πΉβ€˜π‘Œ) ∈ π‘ˆ)
4414, 43sylan2 593 . . . 4 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (πΉβ€˜π‘Œ) ∈ π‘ˆ)
4516, 35, 40, 42, 44setchom 18026 . . 3 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ ((πΉβ€˜π‘‹)(Hom β€˜π‘†)(πΉβ€˜π‘Œ)) = ((πΉβ€˜π‘Œ) ↑m (πΉβ€˜π‘‹)))
4634, 39, 45feq123d 6703 . 2 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ ((π‘‹πΊπ‘Œ):(𝑋(Hom β€˜π‘…)π‘Œ)⟢((πΉβ€˜π‘‹)(Hom β€˜π‘†)(πΉβ€˜π‘Œ)) ↔ ( I β†Ύ (𝑋 RingHom π‘Œ)):(𝑋 RingHom π‘Œ)⟢((πΉβ€˜π‘Œ) ↑m (πΉβ€˜π‘‹))))
4732, 46mpbird 256 1 ((πœ‘ ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡)) β†’ (π‘‹πΊπ‘Œ):(𝑋(Hom β€˜π‘…)π‘Œ)⟢((πΉβ€˜π‘‹)(Hom β€˜π‘†)(πΉβ€˜π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ↦ cmpt 5230   I cid 5572   β†Ύ cres 5677  βŸΆwf 6536  β€“1-1-ontoβ†’wf1o 6539  β€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407   ↑m cmap 8816  WUnicwun 10691  Basecbs 17140  Hom chom 17204  SetCatcsetc 18021   RingHom crh 20240  RingCatALTVcringcALTV 46855
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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
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-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-wun 10693  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-fz 13481  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-plusg 17206  df-hom 17217  df-cco 17218  df-0g 17383  df-setc 18022  df-mhm 18667  df-ghm 19084  df-mgp 19982  df-ur 19999  df-ring 20051  df-rnghom 20243  df-ringcALTV 46857
This theorem is referenced by:  funcringcsetcALTV  46919
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