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Theorem funcestrcsetclem8 16834
Description: Lemma 8 for funcestrcsetc 16836. (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 6212 . . . 4 ( I ↾ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))):((Base‘𝑌) ↑𝑚 (Base‘𝑋))–1-1-onto→((Base‘𝑌) ↑𝑚 (Base‘𝑋))
2 f1of 6175 . . . 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 7921 . . . . 5 (𝑓 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) → 𝑓:(Base‘𝑋)⟶(Base‘𝑌))
5 fvex 6239 . . . . . . . . . 10 (Base‘𝑌) ∈ V
6 fvex 6239 . . . . . . . . . 10 (Base‘𝑋) ∈ V
75, 6pm3.2i 470 . . . . . . . . 9 ((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V)
8 elmapg 7912 . . . . . . . . . 10 (((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V) → (𝑓 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ↔ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)))
98bicomd 213 . . . . . . . . 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 500 . . . . . . 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 16827 . . . . . . . . . 10 ((𝜑𝑌𝐵) → (𝐹𝑌) = (Base‘𝑌))
1918adantrl 752 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑌) = (Base‘𝑌))
2012, 13, 14, 15, 16, 17funcestrcsetclem1 16827 . . . . . . . . . 10 ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
2120adantrr 753 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑋) = (Base‘𝑋))
2219, 21oveq12d 6708 . . . . . . . 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 2733 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → 𝑓 ∈ ((𝐹𝑌) ↑𝑚 (𝐹𝑋)))
2524ex 449 . . . . 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 3642 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ⊆ ((𝐹𝑌) ↑𝑚 (𝐹𝑋)))
283, 27fssd 6095 . 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 2651 . . . 4 (Base‘𝑋) = (Base‘𝑋)
31 eqid 2651 . . . 4 (Base‘𝑌) = (Base‘𝑌)
3212, 13, 14, 15, 16, 17, 29, 30, 31funcestrcsetclem5 16831 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌) = ( I ↾ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))))
3316adantr 480 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑈 ∈ WUni)
34 eqid 2651 . . . 4 (Hom ‘𝐸) = (Hom ‘𝐸)
3512, 16estrcbas 16812 . . . . . . . . 9 (𝜑𝑈 = (Base‘𝐸))
3635, 14syl6reqr 2704 . . . . . . . 8 (𝜑𝐵 = 𝑈)
3736eleq2d 2716 . . . . . . 7 (𝜑 → (𝑋𝐵𝑋𝑈))
3837biimpcd 239 . . . . . 6 (𝑋𝐵 → (𝜑𝑋𝑈))
3938adantr 480 . . . . 5 ((𝑋𝐵𝑌𝐵) → (𝜑𝑋𝑈))
4039impcom 445 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝑈)
4136eleq2d 2716 . . . . . . 7 (𝜑 → (𝑌𝐵𝑌𝑈))
4241biimpd 219 . . . . . 6 (𝜑 → (𝑌𝐵𝑌𝑈))
4342adantld 482 . . . . 5 (𝜑 → ((𝑋𝐵𝑌𝐵) → 𝑌𝑈))
4443imp 444 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝑈)
4512, 33, 34, 40, 44, 30, 31estrchom 16814 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(Hom ‘𝐸)𝑌) = ((Base‘𝑌) ↑𝑚 (Base‘𝑋)))
46 eqid 2651 . . . 4 (Hom ‘𝑆) = (Hom ‘𝑆)
4712, 13, 14, 15, 16, 17funcestrcsetclem2 16828 . . . . 5 ((𝜑𝑋𝐵) → (𝐹𝑋) ∈ 𝑈)
4847adantrr 753 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑋) ∈ 𝑈)
4912, 13, 14, 15, 16, 17funcestrcsetclem2 16828 . . . . 5 ((𝜑𝑌𝐵) → (𝐹𝑌) ∈ 𝑈)
5049adantrl 752 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑌) ∈ 𝑈)
5113, 33, 46, 48, 50setchom 16777 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)) = ((𝐹𝑌) ↑𝑚 (𝐹𝑋)))
5232, 45, 51feq123d 6072 . 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 247 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐸)𝑌)⟶((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cmpt 4762   I cid 5052  cres 5145  wf 5922  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  cmpt2 6692  𝑚 cmap 7899  WUnicwun 9560  Basecbs 15904  Hom chom 15999  SetCatcsetc 16772  ExtStrCatcestrc 16809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-wun 9562  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-hom 16013  df-cco 16014  df-setc 16773  df-estrc 16810
This theorem is referenced by:  funcestrcsetc  16836  fthestrcsetc  16837  fullestrcsetc  16838
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