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Theorem funcestrcsetclem8 17376
Description: Lemma 8 for funcestrcsetc 17378. (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 6625 . . . 4 ( I ↾ ((Base‘𝑌) ↑m (Base‘𝑋))):((Base‘𝑌) ↑m (Base‘𝑋))–1-1-onto→((Base‘𝑌) ↑m (Base‘𝑋))
2 f1of 6588 . . . 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 8403 . . . . 5 (𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) → 𝑓:(Base‘𝑋)⟶(Base‘𝑌))
5 fvex 6656 . . . . . . . . . 10 (Base‘𝑌) ∈ V
6 fvex 6656 . . . . . . . . . 10 (Base‘𝑋) ∈ V
75, 6pm3.2i 474 . . . . . . . . 9 ((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V)
8 elmapg 8394 . . . . . . . . . 10 (((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V) → (𝑓 ∈ ((Base‘𝑌) ↑m (Base‘𝑋)) ↔ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)))
98bicomd 226 . . . . . . . . 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 480 . . . . . . 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 17369 . . . . . . . . . 10 ((𝜑𝑌𝐵) → (𝐹𝑌) = (Base‘𝑌))
1918adantrl 715 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑌) = (Base‘𝑌))
2012, 13, 14, 15, 16, 17funcestrcsetclem1 17369 . . . . . . . . . 10 ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
2120adantrr 716 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑋) = (Base‘𝑋))
2219, 21oveq12d 7148 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((𝐹𝑌) ↑m (𝐹𝑋)) = ((Base‘𝑌) ↑m (Base‘𝑋)))
2322adantr 484 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → ((𝐹𝑌) ↑m (𝐹𝑋)) = ((Base‘𝑌) ↑m (Base‘𝑋)))
2411, 23eleqtrrd 2915 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → 𝑓 ∈ ((𝐹𝑌) ↑m (𝐹𝑋)))
2524ex 416 . . . . 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 3949 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((Base‘𝑌) ↑m (Base‘𝑋)) ⊆ ((𝐹𝑌) ↑m (𝐹𝑋)))
283, 27fssd 6501 . 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 2821 . . . 4 (Base‘𝑋) = (Base‘𝑋)
31 eqid 2821 . . . 4 (Base‘𝑌) = (Base‘𝑌)
3212, 13, 14, 15, 16, 17, 29, 30, 31funcestrcsetclem5 17373 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌) = ( I ↾ ((Base‘𝑌) ↑m (Base‘𝑋))))
3316adantr 484 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑈 ∈ WUni)
34 eqid 2821 . . . 4 (Hom ‘𝐸) = (Hom ‘𝐸)
3512, 16estrcbas 17354 . . . . . . . . 9 (𝜑𝑈 = (Base‘𝐸))
3635, 14syl6reqr 2875 . . . . . . . 8 (𝜑𝐵 = 𝑈)
3736eleq2d 2897 . . . . . . 7 (𝜑 → (𝑋𝐵𝑋𝑈))
3837biimpcd 252 . . . . . 6 (𝑋𝐵 → (𝜑𝑋𝑈))
3938adantr 484 . . . . 5 ((𝑋𝐵𝑌𝐵) → (𝜑𝑋𝑈))
4039impcom 411 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝑈)
4136eleq2d 2897 . . . . . . 7 (𝜑 → (𝑌𝐵𝑌𝑈))
4241biimpd 232 . . . . . 6 (𝜑 → (𝑌𝐵𝑌𝑈))
4342adantld 494 . . . . 5 (𝜑 → ((𝑋𝐵𝑌𝐵) → 𝑌𝑈))
4443imp 410 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝑈)
4512, 33, 34, 40, 44, 30, 31estrchom 17356 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(Hom ‘𝐸)𝑌) = ((Base‘𝑌) ↑m (Base‘𝑋)))
46 eqid 2821 . . . 4 (Hom ‘𝑆) = (Hom ‘𝑆)
4712, 13, 14, 15, 16, 17funcestrcsetclem2 17370 . . . . 5 ((𝜑𝑋𝐵) → (𝐹𝑋) ∈ 𝑈)
4847adantrr 716 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑋) ∈ 𝑈)
4912, 13, 14, 15, 16, 17funcestrcsetclem2 17370 . . . . 5 ((𝜑𝑌𝐵) → (𝐹𝑌) ∈ 𝑈)
5049adantrl 715 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑌) ∈ 𝑈)
5113, 33, 46, 48, 50setchom 17319 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)) = ((𝐹𝑌) ↑m (𝐹𝑋)))
5232, 45, 51feq123d 6476 . 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 260 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐸)𝑌)⟶((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  Vcvv 3471  cmpt 5119   I cid 5432  cres 5530  wf 6324  1-1-ontowf1o 6327  cfv 6328  (class class class)co 7130  cmpo 7132  m cmap 8381  WUnicwun 10099  Basecbs 16462  Hom chom 16555  SetCatcsetc 17314  ExtStrCatcestrc 17351
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-oadd 8081  df-er 8264  df-map 8383  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-wun 10101  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-nn 11616  df-2 11678  df-3 11679  df-4 11680  df-5 11681  df-6 11682  df-7 11683  df-8 11684  df-9 11685  df-n0 11876  df-z 11960  df-dec 12077  df-uz 12222  df-fz 12876  df-struct 16464  df-ndx 16465  df-slot 16466  df-base 16468  df-hom 16568  df-cco 16569  df-setc 17315  df-estrc 17352
This theorem is referenced by:  funcestrcsetc  17378  fthestrcsetc  17379  fullestrcsetc  17380
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