Theorem funcestrcsetclem8 18321

Description: Lemma 8 for funcestrcsetc 18323. (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 6863	. . . 4 ⊢ ( I ↾ ((Base‘𝑌) ↑m (Base‘𝑋))):((Base‘𝑌) ↑m (Base‘𝑋))–1-1-onto→((Base‘𝑌) ↑m (Base‘𝑋))
2		f1of 6824	. . . 4 ⊢ (( I ↾ ((Base‘𝑌) ↑m (Base‘𝑋))):((Base‘𝑌) ↑m (Base‘𝑋))–1-1-onto→((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 8869	. . . . 5 ⊢ (𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) → 𝑓:(Base‘𝑋)⟶(Base‘𝑌))
5		fvex 6898	. . . . . . . . . 10 ⊢ (Base‘𝑌) ∈ V
6		fvex 6898	. . . . . . . . . 10 ⊢ (Base‘𝑋) ∈ V
7	5, 6	pm3.2i 476	. . . . . . . . 9 ⊢ ((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V)
8		elmapg 8859	. . . . . . . . . 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 18314	. . . . . . . . . 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 18314	. . . . . . . . . 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 7438	. . . . . . . 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 2864	. . . . . 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 3937	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((Base‘𝑌) ↑m (Base‘𝑋)) ⊆ ((𝐹‘𝑌) ↑m (𝐹‘𝑋)))
28	3, 27	fssd 6727	. 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 2761	. . . 4 ⊢ (Base‘𝑋) = (Base‘𝑋)
31		eqid 2761	. . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌)
32	12, 13, 14, 15, 16, 17, 29, 30, 31	funcestrcsetclem5 18318	. . 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 2761	. . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
35	12, 16	estrcbas 18299	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 = (Base‘𝐸))
36	14, 35	eqtr4id 2815	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = 𝑈)
37	36	eleq2d 2847	. . . . . . 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 2847	. . . . . . 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 18301	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(Hom ‘𝐸)𝑌) = ((Base‘𝑌) ↑m (Base‘𝑋)))
46		eqid 2761	. . . 4 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
47	12, 13, 14, 15, 16, 17	funcestrcsetclem2 18315	. . . . 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 18315	. . . . 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 18255	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝐹‘𝑋)(Hom ‘𝑆)(𝐹‘𝑌)) = ((𝐹‘𝑌) ↑m (𝐹‘𝑋)))
52	32, 45, 51	feq123d 6698	. 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 2145  Vcvv 3451   ↦ cmpt 5186   I cid 5545   ↾ cres 5653  ⟶wf 6534  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422   ↑_m cmap 8847  WUnicwun 10785  Basecbs 17387  Hom chom 17439  SetCatcsetc 18250  ExtStrCatcestrc 18296

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-wun 10787  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-struct 17325  df-slot 17360  df-ndx 17372  df-base 17388  df-hom 17452  df-cco 17453  df-setc 18251  df-estrc 18297

This theorem is used by:  funcestrcsetc  18323  fthestrcsetc  18324  fullestrcsetc  18325