Theorem funcestrcsetclem8 18371

Description: Lemma 8 for funcestrcsetc 18373. (Contributed by AV, 15-Feb-2020.)

Hypotheses

Ref	Expression
funcestrcsetc.e	⊢ 𝐸 = (ExtStrCat‘𝑈)
funcestrcsetc.s	⊢ 𝑆 = (SetCat‘𝑈)
funcestrcsetc.b	⊢ 𝐵 = (Base‘𝐸)
funcestrcsetc.c	⊢ 𝐶 = (Base‘𝑆)
funcestrcsetc.u	⊢ (𝜑 → 𝑈 ∈ WUni)
funcestrcsetc.f	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))
funcestrcsetc.g	⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))

Assertion

Ref	Expression
funcestrcsetclem8	⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐸)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝑆)(𝐹‘𝑌)))

Distinct variable groups:   𝑥,𝐵   𝑥,𝑋   𝜑,𝑥   𝑥,𝐶   𝑦,𝐵,𝑥   𝑦,𝑋   𝜑,𝑦   𝑥,𝑌,𝑦

Allowed substitution hints:   𝐶(𝑦)   𝑆(𝑥, 𝑦)   𝑈(𝑥, 𝑦)   𝐸(𝑥, 𝑦)   𝐹(𝑥, 𝑦)   𝐺(𝑥, 𝑦)

Proof of Theorem funcestrcsetclem8

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1oi 6894	. . . 4 ⊢ ( I ↾ ((Base‘𝑌) ↑m (Base‘𝑋))):((Base‘𝑌) ↑m (Base‘𝑋))⟶1-1onto((Base‘𝑌) ↑m (Base‘𝑋))
2		f1of 6854	. . . 4 ⊢ (( I ↾ ((Base‘𝑌) ↑m (Base‘𝑋))):((Base‘𝑌) ↑m (Base‘𝑋))⟶1-1onto((Base‘𝑌) ↑m (Base‘𝑋)) → ( I ↾ ((Base‘𝑌) ↑m (Base‘𝑋))):((Base‘𝑌) ↑m (Base‘𝑋))⟶((Base‘𝑌) ↑m (Base‘𝑋)))
3	1, 2	mp1i 14	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ( I ↾ ((Base‘𝑌) ↑m (Base‘𝑋))):((Base‘𝑌) ↑m (Base‘𝑋))⟶((Base‘𝑌) ↑m (Base‘𝑋)))
4		elmapi 8907	. . . . 5 ⊢ (𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) → 𝑓:(Base‘𝑋)⟶(Base‘𝑌))
5		fvex 6929	. . . . . . . . . 10 ⊢ (Base‘𝑌) ∈ V
6		fvex 6929	. . . . . . . . . 10 ⊢ (Base‘𝑋) ∈ V
7	5, 6	pm3.2i 476	. . . . . . . . 9 ⊢ ((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V)
8		elmapg 8897	. . . . . . . . . 10 ⊢ (((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V) → (𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ↔ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)))
9	8	bicomd 226	. . . . . . . . 9 ⊢ (((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V) → (𝑓:(Base‘𝑋)⟶(Base‘𝑌) ↔ 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))))
10	7, 9	mp1i 14	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑓:(Base‘𝑋)⟶(Base‘𝑌) ↔ 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))))
11	10	biimpa 482	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)))
12		funcestrcsetc.e	. . . . . . . . . . 11 ⊢ 𝐸 = (ExtStrCat‘𝑈)
13		funcestrcsetc.s	. . . . . . . . . . 11 ⊢ 𝑆 = (SetCat‘𝑈)
14		funcestrcsetc.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐸)
15		funcestrcsetc.c	. . . . . . . . . . 11 ⊢ 𝐶 = (Base‘𝑆)
16		funcestrcsetc.u	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ WUni)
17		funcestrcsetc.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))
18	12, 13, 14, 15, 16, 17	funcestrcsetclem1 18364	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) = (Base‘𝑌))
19	18	adantrl 729	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹‘𝑌) = (Base‘𝑌))
20	12, 13, 14, 15, 16, 17	funcestrcsetclem1 18364	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋))
21	20	adantrr 730	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹‘𝑋) = (Base‘𝑋))
22	19, 21	oveq12d 7469	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝐹‘𝑌) ↑m (𝐹‘𝑋)) = ((Base‘𝑌) ↑m (Base‘𝑋)))
23	22	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → ((𝐹‘𝑌) ↑m (𝐹‘𝑋)) = ((Base‘𝑌) ↑m (Base‘𝑋)))
24	11, 23	eleqtrrd 2867	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → 𝑓 ∈ ((𝐹‘𝑌) ↑m (𝐹‘𝑋)))
25	24	ex 418	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑓:(Base‘𝑋)⟶(Base‘𝑌) → 𝑓 ∈ ((𝐹‘𝑌) ↑m (𝐹‘𝑋))))
26	4, 25	syl5 35	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) → 𝑓 ∈ ((𝐹‘𝑌) ↑m (𝐹‘𝑋))))
27	26	ssrdv 3919	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((Base‘𝑌) ↑m (Base‘𝑋)) ⊆ ((𝐹‘𝑌) ↑m (𝐹‘𝑋)))
28	3, 27	fssd 6757	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ( I ↾ ((Base‘𝑌) ↑m (Base‘𝑋))):((Base‘𝑌) ↑m (Base‘𝑋))⟶((𝐹‘𝑌) ↑m (𝐹‘𝑋)))
29		funcestrcsetc.g	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
30		eqid 2763	. . . 4 ⊢ (Base‘𝑋) = (Base‘𝑋)
31		eqid 2763	. . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌)
32	12, 13, 14, 15, 16, 17, 29, 30, 31	funcestrcsetclem5 18368	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌) = ( I ↾ ((Base‘𝑌) ↑m (Base‘𝑋))))
33	16	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑈 ∈ WUni)
34		eqid 2763	. . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
35	12, 16	estrcbas 18349	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 = (Base‘𝐸))
36	14, 35	eqtr4id 2817	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = 𝑈)
37	36	eleq2d 2850	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ 𝑈))
38	37	biimpcd 252	. . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (𝜑 → 𝑋 ∈ 𝑈))
39	38	adantr 486	. . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝜑 → 𝑋 ∈ 𝑈))
40	39	impcom 413	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝑈)
41	36	eleq2d 2850	. . . . . . 7 ⊢ (𝜑 → (𝑌 ∈ 𝐵 ↔ 𝑌 ∈ 𝑈))
42	41	biimpd 232	. . . . . 6 ⊢ (𝜑 → (𝑌 ∈ 𝐵 → 𝑌 ∈ 𝑈))
43	42	adantld 496	. . . . 5 ⊢ (𝜑 → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝑈))
44	43	imp 412	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝑈)
45	12, 33, 34, 40, 44, 30, 31	estrchom 18351	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(Hom ‘𝐸)𝑌) = ((Base‘𝑌) ↑m (Base‘𝑋)))
46		eqid 2763	. . . 4 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
47	12, 13, 14, 15, 16, 17	funcestrcsetclem2 18365	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝑈)
48	47	adantrr 730	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹‘𝑋) ∈ 𝑈)
49	12, 13, 14, 15, 16, 17	funcestrcsetclem2 18365	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) ∈ 𝑈)
50	49	adantrl 729	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹‘𝑌) ∈ 𝑈)
51	13, 33, 46, 48, 50	setchom 18305	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝐹‘𝑋)(Hom ‘𝑆)(𝐹‘𝑌)) = ((𝐹‘𝑌) ↑m (𝐹‘𝑋)))
52	32, 45, 51	feq123d 6728	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋𝐺𝑌):(𝑋(Hom ‘𝐸)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝑆)(𝐹‘𝑌)) ↔ ( I ↾ ((Base‘𝑌) ↑m (Base‘𝑋))):((Base‘𝑌) ↑m (Base‘𝑋))⟶((𝐹‘𝑌) ↑m (𝐹‘𝑋))))
53	28, 52	mpbird 260	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐸)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝑆)(𝐹‘𝑌)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ↦ cmpt 5192   I cid 5552   ↾ cres 5660  ⟶wf 6562  ⟶1-1ontowf1o 6565  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453   ↑_m cmap 8885  WUnicwun 10824  Basecbs 17431  Hom chom 17483  SetCatcsetc 18300  ExtStrCatcestrc 18346

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-wun 10826  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-7 12450  df-8 12451  df-9 12452  df-n0 12647  df-z 12734  df-dec 12855  df-uz 13006  df-fz 13680  df-struct 17369  df-slot 17404  df-ndx 17416  df-base 17432  df-hom 17497  df-cco 17498  df-setc 18301  df-estrc 18347

This theorem is used by:  funcestrcsetc  18373  fthestrcsetc  18374  fullestrcsetc  18375