Theorem funcringcsetcALTV2lem8 44388

Description: Lemma 8 for funcringcsetcALTV2 44390. (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 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))
3	1, 2	mp1i 13	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ( I ↾ (𝑋 RingHom 𝑌)):(𝑋 RingHom 𝑌)⟶(𝑋 RingHom 𝑌))
4		eqid 2820	. . . . . 6 ⊢ (Base‘𝑋) = (Base‘𝑋)
5		eqid 2820	. . . . . 6 ⊢ (Base‘𝑌) = (Base‘𝑌)
6	4, 5	rhmf 19474	. . . . 5 ⊢ (𝑓 ∈ (𝑋 RingHom 𝑌) → 𝑓:(Base‘𝑋)⟶(Base‘𝑌))
7		fvex 6680	. . . . . . . . . 10 ⊢ (Base‘𝑌) ∈ V
8		fvex 6680	. . . . . . . . . 10 ⊢ (Base‘𝑋) ∈ V
9	7, 8	pm3.2i 473	. . . . . . . . 9 ⊢ ((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V)
10		elmapg 8416	. . . . . . . . . 10 ⊢ (((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V) → (𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ↔ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)))
11	10	bicomd 225	. . . . . . . . 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 479	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)))
14		simpr 487	. . . . . . . . . 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 44381	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) = (Base‘𝑌))
22	14, 21	sylan2 594	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹‘𝑌) = (Base‘𝑌))
23		simpl 485	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐵)
24	15, 16, 17, 18, 19, 20	funcringcsetcALTV2lem1 44381	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋))
25	23, 24	sylan2 594	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹‘𝑋) = (Base‘𝑋))
26	22, 25	oveq12d 7171	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝐹‘𝑌) ↑m (𝐹‘𝑋)) = ((Base‘𝑌) ↑m (Base‘𝑋)))
27	26	adantr 483	. . . . . . 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 2915	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → 𝑓 ∈ ((𝐹‘𝑌) ↑m (𝐹‘𝑋)))
29	28	ex 415	. . . . 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 3970	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 RingHom 𝑌) ⊆ ((𝐹‘𝑌) ↑m (𝐹‘𝑋)))
32	3, 31	fssd 6525	. 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 44385	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑋 RingHom 𝑌)))
35	19	adantr 483	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑈 ∈ WUni)
36		eqid 2820	. . . 4 ⊢ (Hom ‘𝑅) = (Hom ‘𝑅)
37	23	adantl 484	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
38	14	adantl 484	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
39	15, 17, 35, 36, 37, 38	ringchom 44358	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(Hom ‘𝑅)𝑌) = (𝑋 RingHom 𝑌))
40		eqid 2820	. . . 4 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
41	15, 16, 17, 18, 19, 20	funcringcsetcALTV2lem2 44382	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝑈)
42	23, 41	sylan2 594	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹‘𝑋) ∈ 𝑈)
43	15, 16, 17, 18, 19, 20	funcringcsetcALTV2lem2 44382	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) ∈ 𝑈)
44	14, 43	sylan2 594	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹‘𝑌) ∈ 𝑈)
45	16, 35, 40, 42, 44	setchom 17336	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝐹‘𝑋)(Hom ‘𝑆)(𝐹‘𝑌)) = ((𝐹‘𝑌) ↑m (𝐹‘𝑋)))
46	34, 39, 45	feq123d 6500	. 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 259	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝑅)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝑆)(𝐹‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1536   ∈ wcel 2113  Vcvv 3493   ↦ cmpt 5143   I cid 5456   ↾ cres 5554  ⟶wf 6348  –1-1-onto→wf1o 6351  ‘cfv 6352  (class class class)co 7153   ∈ cmpo 7155   ↑_m cmap 8403  WUnicwun 10119  Basecbs 16479  Hom chom 16572  SetCatcsetc 17331   RingHom crh 19460  RingCatcringc 44348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5327  ax-un 7458  ax-cnex 10590  ax-resscn 10591  ax-1cn 10592  ax-icn 10593  ax-addcl 10594  ax-addrcl 10595  ax-mulcl 10596  ax-mulrcl 10597  ax-mulcom 10598  ax-addass 10599  ax-mulass 10600  ax-distr 10601  ax-i2m1 10602  ax-1ne0 10603  ax-1rid 10604  ax-rnegex 10605  ax-rrecex 10606  ax-cnre 10607  ax-pre-lttri 10608  ax-pre-lttrn 10609  ax-pre-ltadd 10610  ax-pre-mulgt0 10611

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-nel 3123  df-ral 3142  df-rex 3143  df-reu 3144  df-rab 3146  df-v 3495  df-sbc 3771  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4465  df-pw 4538  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4836  df-int 4874  df-iun 4918  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5457  df-eprel 5462  df-po 5471  df-so 5472  df-fr 5511  df-we 5513  df-xp 5558  df-rel 5559  df-cnv 5560  df-co 5561  df-dm 5562  df-rn 5563  df-res 5564  df-ima 5565  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7111  df-ov 7156  df-oprab 7157  df-mpo 7158  df-om 7578  df-1st 7686  df-2nd 7687  df-wrecs 7944  df-recs 8005  df-rdg 8043  df-1o 8099  df-oadd 8103  df-er 8286  df-map 8405  df-en 8507  df-dom 8508  df-sdom 8509  df-fin 8510  df-wun 10121  df-pnf 10674  df-mnf 10675  df-xr 10676  df-ltxr 10677  df-le 10678  df-sub 10869  df-neg 10870  df-nn 11636  df-2 11698  df-3 11699  df-4 11700  df-5 11701  df-6 11702  df-7 11703  df-8 11704  df-9 11705  df-n0 11896  df-z 11980  df-dec 12097  df-uz 12242  df-fz 12891  df-struct 16481  df-ndx 16482  df-slot 16483  df-base 16485  df-sets 16486  df-ress 16487  df-plusg 16574  df-hom 16585  df-cco 16586  df-0g 16711  df-resc 17077  df-setc 17332  df-estrc 17369  df-mhm 17952  df-ghm 18352  df-mgp 19236  df-ur 19248  df-ring 19295  df-rnghom 19463  df-ringc 44350

This theorem is referenced by:  funcringcsetcALTV2  44390