Theorem funcestrcsetclem8 18113

Description: Lemma 8 for funcestrcsetc 18115. (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 6818	. . . 4 ⊢ ( I ↾ ((Base‘𝑌) ↑m (Base‘𝑋))):((Base‘𝑌) ↑m (Base‘𝑋))–1-1-onto→((Base‘𝑌) ↑m (Base‘𝑋))
2		f1of 6780	. . . 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 13	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ( I ↾ ((Base‘𝑌) ↑m (Base‘𝑋))):((Base‘𝑌) ↑m (Base‘𝑋))⟶((Base‘𝑌) ↑m (Base‘𝑋)))
4		elmapi 8796	. . . . 5 ⊢ (𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) → 𝑓:(Base‘𝑋)⟶(Base‘𝑌))
5		fvex 6853	. . . . . . . . . 10 ⊢ (Base‘𝑌) ∈ V
6		fvex 6853	. . . . . . . . . 10 ⊢ (Base‘𝑋) ∈ V
7	5, 6	pm3.2i 470	. . . . . . . . 9 ⊢ ((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V)
8		elmapg 8786	. . . . . . . . . 10 ⊢ (((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V) → (𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ↔ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)))
9	8	bicomd 223	. . . . . . . . 9 ⊢ (((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V) → (𝑓:(Base‘𝑋)⟶(Base‘𝑌) ↔ 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))))
10	7, 9	mp1i 13	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑓:(Base‘𝑋)⟶(Base‘𝑌) ↔ 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))))
11	10	biimpa 476	. . . . . . 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 18106	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) = (Base‘𝑌))
19	18	adantrl 717	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹‘𝑌) = (Base‘𝑌))
20	12, 13, 14, 15, 16, 17	funcestrcsetclem1 18106	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋))
21	20	adantrr 718	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹‘𝑋) = (Base‘𝑋))
22	19, 21	oveq12d 7385	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝐹‘𝑌) ↑m (𝐹‘𝑋)) = ((Base‘𝑌) ↑m (Base‘𝑋)))
23	22	adantr 480	. . . . . . 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 2839	. . . . . 6 ⊢ (((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → 𝑓 ∈ ((𝐹‘𝑌) ↑m (𝐹‘𝑋)))
25	24	ex 412	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑓:(Base‘𝑋)⟶(Base‘𝑌) → 𝑓 ∈ ((𝐹‘𝑌) ↑m (𝐹‘𝑋))))
26	4, 25	syl5 34	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) → 𝑓 ∈ ((𝐹‘𝑌) ↑m (𝐹‘𝑋))))
27	26	ssrdv 3927	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((Base‘𝑌) ↑m (Base‘𝑋)) ⊆ ((𝐹‘𝑌) ↑m (𝐹‘𝑋)))
28	3, 27	fssd 6685	. 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 2736	. . . 4 ⊢ (Base‘𝑋) = (Base‘𝑋)
31		eqid 2736	. . . 4 ⊢ (Base‘𝑌) = (Base‘𝑌)
32	12, 13, 14, 15, 16, 17, 29, 30, 31	funcestrcsetclem5 18110	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌) = ( I ↾ ((Base‘𝑌) ↑m (Base‘𝑋))))
33	16	adantr 480	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑈 ∈ WUni)
34		eqid 2736	. . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
35	12, 16	estrcbas 18091	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 = (Base‘𝐸))
36	14, 35	eqtr4id 2790	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = 𝑈)
37	36	eleq2d 2822	. . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ↔ 𝑋 ∈ 𝑈))
38	37	biimpcd 249	. . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (𝜑 → 𝑋 ∈ 𝑈))
39	38	adantr 480	. . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝜑 → 𝑋 ∈ 𝑈))
40	39	impcom 407	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝑈)
41	36	eleq2d 2822	. . . . . . 7 ⊢ (𝜑 → (𝑌 ∈ 𝐵 ↔ 𝑌 ∈ 𝑈))
42	41	biimpd 229	. . . . . 6 ⊢ (𝜑 → (𝑌 ∈ 𝐵 → 𝑌 ∈ 𝑈))
43	42	adantld 490	. . . . 5 ⊢ (𝜑 → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝑈))
44	43	imp 406	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝑈)
45	12, 33, 34, 40, 44, 30, 31	estrchom 18093	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋(Hom ‘𝐸)𝑌) = ((Base‘𝑌) ↑m (Base‘𝑋)))
46		eqid 2736	. . . 4 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
47	12, 13, 14, 15, 16, 17	funcestrcsetclem2 18107	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝑈)
48	47	adantrr 718	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹‘𝑋) ∈ 𝑈)
49	12, 13, 14, 15, 16, 17	funcestrcsetclem2 18107	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐵) → (𝐹‘𝑌) ∈ 𝑈)
50	49	adantrl 717	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐹‘𝑌) ∈ 𝑈)
51	13, 33, 46, 48, 50	setchom 18047	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝐹‘𝑋)(Hom ‘𝑆)(𝐹‘𝑌)) = ((𝐹‘𝑌) ↑m (𝐹‘𝑋)))
52	32, 45, 51	feq123d 6657	. 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 257	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐸)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝑆)(𝐹‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ↦ cmpt 5166   I cid 5525   ↾ cres 5633  ⟶wf 6494  –1-1-onto→wf1o 6497  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369   ↑_m cmap 8773  WUnicwun 10623  Basecbs 17179  Hom chom 17231  SetCatcsetc 18042  ExtStrCatcestrc 18088

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-wun 10625  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-fz 13462  df-struct 17117  df-slot 17152  df-ndx 17164  df-base 17180  df-hom 17244  df-cco 17245  df-setc 18043  df-estrc 18089

This theorem is referenced by:  funcestrcsetc  18115  fthestrcsetc  18116  fullestrcsetc  18117