Theorem funcringcsetcALTV2lem8 43812

Description: Lemma 8 for funcringcsetcALTV2 43814. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)

Hypotheses

Ref	Expression
funcringcsetcALTV2.r	⊢ 𝑅 = (RingCat‘𝑈)
funcringcsetcALTV2.s	⊢ 𝑆 = (SetCat‘𝑈)
funcringcsetcALTV2.b	⊢ 𝐵 = (Base‘𝑅)
funcringcsetcALTV2.c	⊢ 𝐶 = (Base‘𝑆)
funcringcsetcALTV2.u	⊢ (𝜑 → 𝑈 ∈ WUni)
funcringcsetcALTV2.f	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))
funcringcsetcALTV2.g	⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))

Assertion

Ref	Expression
funcringcsetcALTV2lem8	⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝑅)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝑆)(𝐹‘𝑌)))

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 funcringcsetcALTV2lem8

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1oi 6520	. . . 4 ⊢ ( I ↾ (𝑋 RingHom 𝑌)):(𝑋 RingHom 𝑌)–1-1-onto→(𝑋 RingHom 𝑌)
2		f1of 6483	. . . 4 ⊢ (( I ↾ (𝑋 RingHom 𝑌)):(𝑋 RingHom 𝑌)–1-1-onto→(𝑋 RingHom 𝑌) → ( I ↾ (𝑋 RingHom 𝑌)):(𝑋 RingHom 𝑌)⟶(𝑋 RingHom 𝑌))
3	1, 2	mp1i 13	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ( I ↾ (𝑋 RingHom 𝑌)):(𝑋 RingHom 𝑌)⟶(𝑋 RingHom 𝑌))
4		eqid 2795	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
5		eqid 2795	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘𝑌)
6	4, 5	rhmf 19168	. . . . 5 ⊢ (𝑓 ∈ (𝑋 RingHom 𝑌) → 𝑓:(Base‘𝑋)⟶(Base‘𝑌))
7		fvex 6551	. . . . . . . . . 10 ⊢ (Base‘𝑌) ∈ V
8		fvex 6551	. . . . . . . . . 10 ⊢ (Base‘𝑋) ∈ V
9	7, 8	pm3.2i 471	. . . . . . . . 9 ⊢ ((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V)
10		elmapg 8269	. . . . . . . . . 10 ⊢ (((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V) → (𝑓 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ↔ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)))
11	10	bicomd 224	. . . . . . . . 9 ⊢ (((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V) → (𝑓:(Base‘𝑋)⟶(Base‘𝑌) ↔ 𝑓 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))))
12	9, 11	mp1i 13	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑓:(Base‘𝑋)⟶(Base‘𝑌) ↔ 𝑓 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))))
13	12	biimpa 477	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → 𝑓 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)))
14		simpr 485	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
15		funcringcsetcALTV2.r	. . . . . . . . . . 11 ⊢ 𝑅 = (RingCat‘𝑈)
16		funcringcsetcALTV2.s	. . . . . . . . . . 11 ⊢ 𝑆 = (SetCat‘𝑈)
17		funcringcsetcALTV2.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑅)
18		funcringcsetcALTV2.c	. . . . . . . . . . 11 ⊢ 𝐶 = (Base‘𝑆)
19		funcringcsetcALTV2.u	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ WUni)
20		funcringcsetcALTV2.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))
21	15, 16, 17, 18, 19, 20	funcringcsetcALTV2lem1 43805	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) = (Base‘𝑌))
22	14, 21	sylan2 592	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹‘𝑌) = (Base‘𝑌))
23		simpl 483	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
24	15, 16, 17, 18, 19, 20	funcringcsetcALTV2lem1 43805	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋))
25	23, 24	sylan2 592	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹‘𝑋) = (Base‘𝑋))
26	22, 25	oveq12d 7034	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝐹‘𝑌) ↑𝑚 (𝐹‘𝑋)) = ((Base‘𝑌) ↑𝑚 (Base‘𝑋)))
27	26	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → ((𝐹‘𝑌) ↑𝑚 (𝐹‘𝑋)) = ((Base‘𝑌) ↑𝑚 (Base‘𝑋)))
28	13, 27	eleqtrrd 2886	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → 𝑓 ∈ ((𝐹‘𝑌) ↑𝑚 (𝐹‘𝑋)))
29	28	ex 413	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑓:(Base‘𝑋)⟶(Base‘𝑌) → 𝑓 ∈ ((𝐹‘𝑌) ↑𝑚 (𝐹‘𝑋))))
30	6, 29	syl5 34	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑓 ∈ (𝑋 RingHom 𝑌) → 𝑓 ∈ ((𝐹‘𝑌) ↑𝑚 (𝐹‘𝑋))))
31	30	ssrdv 3895	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 RingHom 𝑌) ⊆ ((𝐹‘𝑌) ↑𝑚 (𝐹‘𝑋)))
32	3, 31	fssd 6396	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ( I ↾ (𝑋 RingHom 𝑌)):(𝑋 RingHom 𝑌)⟶((𝐹‘𝑌) ↑𝑚 (𝐹‘𝑋)))
33		funcringcsetcALTV2.g	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
34	15, 16, 17, 18, 19, 20, 33	funcringcsetcALTV2lem5 43809	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑋 RingHom 𝑌)))
35	19	adantr 481	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑈 ∈ WUni)
36		eqid 2795	. . . 4 ⊢ (Hom ‘𝑅) = (Hom ‘𝑅)
37	23	adantl 482	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
38	14	adantl 482	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
39	15, 17, 35, 36, 37, 38	ringchom 43782	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(Hom ‘𝑅)𝑌) = (𝑋 RingHom 𝑌))
40		eqid 2795	. . . 4 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
41	15, 16, 17, 18, 19, 20	funcringcsetcALTV2lem2 43806	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝑈)
42	23, 41	sylan2 592	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹‘𝑋) ∈ 𝑈)
43	15, 16, 17, 18, 19, 20	funcringcsetcALTV2lem2 43806	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) ∈ 𝑈)
44	14, 43	sylan2 592	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹‘𝑌) ∈ 𝑈)
45	16, 35, 40, 42, 44	setchom 17169	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝐹‘𝑋)(Hom ‘𝑆)(𝐹‘𝑌)) = ((𝐹‘𝑌) ↑𝑚 (𝐹‘𝑋)))
46	34, 39, 45	feq123d 6371	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋𝐺𝑌):(𝑋(Hom ‘𝑅)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝑆)(𝐹‘𝑌)) ↔ ( I ↾ (𝑋 RingHom 𝑌)):(𝑋 RingHom 𝑌)⟶((𝐹‘𝑌) ↑𝑚 (𝐹‘𝑋))))
47	32, 46	mpbird 258	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝑅)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝑆)(𝐹‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1522   ∈ wcel 2081  Vcvv 3437   ↦ cmpt 5041   I cid 5347   ↾ cres 5445  ⟶wf 6221  –1-1-onto→wf1o 6224  ‘cfv 6225  (class class class)co 7016   ∈ cmpo 7018   ↑_𝑚 cmap 8256  WUnicwun 9968  Basecbs 16312  Hom chom 16405  SetCatcsetc 17164   RingHom crh 19154  RingCatcringc 43772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-er 8139  df-map 8258  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-wun 9970  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-2 11548  df-3 11549  df-4 11550  df-5 11551  df-6 11552  df-7 11553  df-8 11554  df-9 11555  df-n0 11746  df-z 11830  df-dec 11948  df-uz 12094  df-fz 12743  df-struct 16314  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-ress 16320  df-plusg 16407  df-hom 16418  df-cco 16419  df-0g 16544  df-resc 16910  df-setc 17165  df-estrc 17202  df-mhm 17774  df-ghm 18097  df-mgp 18930  df-ur 18942  df-ring 18989  df-rnghom 19157  df-ringc 43774

This theorem is referenced by:  funcringcsetcALTV2  43814