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Theorem funcestrcsetclem8 18192
Description: Lemma 8 for funcestrcsetc 18194. (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 (𝜑𝑈 ∈ WUni)
funcestrcsetc.f (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
funcestrcsetc.g (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
Assertion
Ref Expression
funcestrcsetclem8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐸)𝑌)⟶((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)))
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.
StepHypRef Expression
1 f1oi 6886 . . . 4 ( I ↾ ((Base‘𝑌) ↑m (Base‘𝑋))):((Base‘𝑌) ↑m (Base‘𝑋))–1-1-onto→((Base‘𝑌) ↑m (Base‘𝑋))
2 f1of 6848 . . . 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)))
31, 2mp1i 13 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ( I ↾ ((Base‘𝑌) ↑m (Base‘𝑋))):((Base‘𝑌) ↑m (Base‘𝑋))⟶((Base‘𝑌) ↑m (Base‘𝑋)))
4 elmapi 8889 . . . . 5 (𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) → 𝑓:(Base‘𝑋)⟶(Base‘𝑌))
5 fvex 6919 . . . . . . . . . 10 (Base‘𝑌) ∈ V
6 fvex 6919 . . . . . . . . . 10 (Base‘𝑋) ∈ V
75, 6pm3.2i 470 . . . . . . . . 9 ((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V)
8 elmapg 8879 . . . . . . . . . 10 (((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V) → (𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ↔ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)))
98bicomd 223 . . . . . . . . 9 (((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V) → (𝑓:(Base‘𝑋)⟶(Base‘𝑌) ↔ 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))))
107, 9mp1i 13 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑓:(Base‘𝑋)⟶(Base‘𝑌) ↔ 𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋))))
1110biimpa 476 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 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 (𝜑𝑈 ∈ WUni)
17 funcestrcsetc.f . . . . . . . . . . 11 (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))
1812, 13, 14, 15, 16, 17funcestrcsetclem1 18185 . . . . . . . . . 10 ((𝜑𝑌𝐵) → (𝐹𝑌) = (Base‘𝑌))
1918adantrl 716 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑌) = (Base‘𝑌))
2012, 13, 14, 15, 16, 17funcestrcsetclem1 18185 . . . . . . . . . 10 ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
2120adantrr 717 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑋) = (Base‘𝑋))
2219, 21oveq12d 7449 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((𝐹𝑌) ↑m (𝐹𝑋)) = ((Base‘𝑌) ↑m (Base‘𝑋)))
2322adantr 480 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → ((𝐹𝑌) ↑m (𝐹𝑋)) = ((Base‘𝑌) ↑m (Base‘𝑋)))
2411, 23eleqtrrd 2844 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → 𝑓 ∈ ((𝐹𝑌) ↑m (𝐹𝑋)))
2524ex 412 . . . . 5 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑓:(Base‘𝑋)⟶(Base‘𝑌) → 𝑓 ∈ ((𝐹𝑌) ↑m (𝐹𝑋))))
264, 25syl5 34 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) → 𝑓 ∈ ((𝐹𝑌) ↑m (𝐹𝑋))))
2726ssrdv 3989 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((Base‘𝑌) ↑m (Base‘𝑋)) ⊆ ((𝐹𝑌) ↑m (𝐹𝑋)))
283, 27fssd 6753 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ( I ↾ ((Base‘𝑌) ↑m (Base‘𝑋))):((Base‘𝑌) ↑m (Base‘𝑋))⟶((𝐹𝑌) ↑m (𝐹𝑋)))
29 funcestrcsetc.g . . . 4 (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
30 eqid 2737 . . . 4 (Base‘𝑋) = (Base‘𝑋)
31 eqid 2737 . . . 4 (Base‘𝑌) = (Base‘𝑌)
3212, 13, 14, 15, 16, 17, 29, 30, 31funcestrcsetclem5 18189 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌) = ( I ↾ ((Base‘𝑌) ↑m (Base‘𝑋))))
3316adantr 480 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑈 ∈ WUni)
34 eqid 2737 . . . 4 (Hom ‘𝐸) = (Hom ‘𝐸)
3512, 16estrcbas 18169 . . . . . . . . 9 (𝜑𝑈 = (Base‘𝐸))
3614, 35eqtr4id 2796 . . . . . . . 8 (𝜑𝐵 = 𝑈)
3736eleq2d 2827 . . . . . . 7 (𝜑 → (𝑋𝐵𝑋𝑈))
3837biimpcd 249 . . . . . 6 (𝑋𝐵 → (𝜑𝑋𝑈))
3938adantr 480 . . . . 5 ((𝑋𝐵𝑌𝐵) → (𝜑𝑋𝑈))
4039impcom 407 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝑈)
4136eleq2d 2827 . . . . . . 7 (𝜑 → (𝑌𝐵𝑌𝑈))
4241biimpd 229 . . . . . 6 (𝜑 → (𝑌𝐵𝑌𝑈))
4342adantld 490 . . . . 5 (𝜑 → ((𝑋𝐵𝑌𝐵) → 𝑌𝑈))
4443imp 406 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝑈)
4512, 33, 34, 40, 44, 30, 31estrchom 18171 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(Hom ‘𝐸)𝑌) = ((Base‘𝑌) ↑m (Base‘𝑋)))
46 eqid 2737 . . . 4 (Hom ‘𝑆) = (Hom ‘𝑆)
4712, 13, 14, 15, 16, 17funcestrcsetclem2 18186 . . . . 5 ((𝜑𝑋𝐵) → (𝐹𝑋) ∈ 𝑈)
4847adantrr 717 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑋) ∈ 𝑈)
4912, 13, 14, 15, 16, 17funcestrcsetclem2 18186 . . . . 5 ((𝜑𝑌𝐵) → (𝐹𝑌) ∈ 𝑈)
5049adantrl 716 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑌) ∈ 𝑈)
5113, 33, 46, 48, 50setchom 18125 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)) = ((𝐹𝑌) ↑m (𝐹𝑋)))
5232, 45, 51feq123d 6725 . 2 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((𝑋𝐺𝑌):(𝑋(Hom ‘𝐸)𝑌)⟶((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)) ↔ ( I ↾ ((Base‘𝑌) ↑m (Base‘𝑋))):((Base‘𝑌) ↑m (Base‘𝑋))⟶((𝐹𝑌) ↑m (𝐹𝑋))))
5328, 52mpbird 257 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐸)𝑌)⟶((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cmpt 5225   I cid 5577  cres 5687  wf 6557  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  cmpo 7433  m cmap 8866  WUnicwun 10740  Basecbs 17247  Hom chom 17308  SetCatcsetc 18120  ExtStrCatcestrc 18166
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-wun 10742  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-hom 17321  df-cco 17322  df-setc 18121  df-estrc 18167
This theorem is referenced by:  funcestrcsetc  18194  fthestrcsetc  18195  fullestrcsetc  18196
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