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Theorem funcestrcsetclem8 16719
Description: Lemma 8 for funcestrcsetc 16721. (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 6136 . . . 4 ( I ↾ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))):((Base‘𝑌) ↑𝑚 (Base‘𝑋))–1-1-onto→((Base‘𝑌) ↑𝑚 (Base‘𝑋))
2 f1of 6099 . . . 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 7831 . . . . 5 (𝑓 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) → 𝑓:(Base‘𝑋)⟶(Base‘𝑌))
5 fvex 6163 . . . . . . . . . 10 (Base‘𝑌) ∈ V
6 fvex 6163 . . . . . . . . . 10 (Base‘𝑋) ∈ V
75, 6pm3.2i 471 . . . . . . . . 9 ((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V)
8 elmapg 7822 . . . . . . . . . 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 501 . . . . . . 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 16712 . . . . . . . . . 10 ((𝜑𝑌𝐵) → (𝐹𝑌) = (Base‘𝑌))
1918adantrl 751 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑌) = (Base‘𝑌))
2012, 13, 14, 15, 16, 17funcestrcsetclem1 16712 . . . . . . . . . 10 ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
2120adantrr 752 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑋) = (Base‘𝑋))
2219, 21oveq12d 6628 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((𝐹𝑌) ↑𝑚 (𝐹𝑋)) = ((Base‘𝑌) ↑𝑚 (Base‘𝑋)))
2322adantr 481 . . . . . . 7 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → ((𝐹𝑌) ↑𝑚 (𝐹𝑋)) = ((Base‘𝑌) ↑𝑚 (Base‘𝑋)))
2411, 23eleqtrrd 2701 . . . . . 6 (((𝜑 ∧ (𝑋𝐵𝑌𝐵)) ∧ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)) → 𝑓 ∈ ((𝐹𝑌) ↑𝑚 (𝐹𝑋)))
2524ex 450 . . . . 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 3593 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ⊆ ((𝐹𝑌) ↑𝑚 (𝐹𝑋)))
283, 27fssd 6019 . 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 2621 . . . 4 (Base‘𝑋) = (Base‘𝑋)
31 eqid 2621 . . . 4 (Base‘𝑌) = (Base‘𝑌)
3212, 13, 14, 15, 16, 17, 29, 30, 31funcestrcsetclem5 16716 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌) = ( I ↾ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))))
3316adantr 481 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑈 ∈ WUni)
34 eqid 2621 . . . 4 (Hom ‘𝐸) = (Hom ‘𝐸)
3512, 16estrcbas 16697 . . . . . . . . 9 (𝜑𝑈 = (Base‘𝐸))
3635, 14syl6reqr 2674 . . . . . . . 8 (𝜑𝐵 = 𝑈)
3736eleq2d 2684 . . . . . . 7 (𝜑 → (𝑋𝐵𝑋𝑈))
3837biimpcd 239 . . . . . 6 (𝑋𝐵 → (𝜑𝑋𝑈))
3938adantr 481 . . . . 5 ((𝑋𝐵𝑌𝐵) → (𝜑𝑋𝑈))
4039impcom 446 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝑈)
4136eleq2d 2684 . . . . . . 7 (𝜑 → (𝑌𝐵𝑌𝑈))
4241biimpd 219 . . . . . 6 (𝜑 → (𝑌𝐵𝑌𝑈))
4342adantld 483 . . . . 5 (𝜑 → ((𝑋𝐵𝑌𝐵) → 𝑌𝑈))
4443imp 445 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝑈)
4512, 33, 34, 40, 44, 30, 31estrchom 16699 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(Hom ‘𝐸)𝑌) = ((Base‘𝑌) ↑𝑚 (Base‘𝑋)))
46 eqid 2621 . . . 4 (Hom ‘𝑆) = (Hom ‘𝑆)
4712, 13, 14, 15, 16, 17funcestrcsetclem2 16713 . . . . 5 ((𝜑𝑋𝐵) → (𝐹𝑋) ∈ 𝑈)
4847adantrr 752 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑋) ∈ 𝑈)
4912, 13, 14, 15, 16, 17funcestrcsetclem2 16713 . . . . 5 ((𝜑𝑌𝐵) → (𝐹𝑌) ∈ 𝑈)
5049adantrl 751 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑌) ∈ 𝑈)
5113, 33, 46, 48, 50setchom 16662 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)) = ((𝐹𝑌) ↑𝑚 (𝐹𝑋)))
5232, 45, 51feq123d 5996 . 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 384   = wceq 1480  wcel 1987  Vcvv 3189  cmpt 4678   I cid 4989  cres 5081  wf 5848  1-1-ontowf1o 5851  cfv 5852  (class class class)co 6610  cmpt2 6612  𝑚 cmap 7809  WUnicwun 9474  Basecbs 15792  Hom chom 15884  SetCatcsetc 16657  ExtStrCatcestrc 16694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-wun 9476  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-3 11032  df-4 11033  df-5 11034  df-6 11035  df-7 11036  df-8 11037  df-9 11038  df-n0 11245  df-z 11330  df-dec 11446  df-uz 11640  df-fz 12277  df-struct 15794  df-ndx 15795  df-slot 15796  df-base 15797  df-hom 15898  df-cco 15899  df-setc 16658  df-estrc 16695
This theorem is referenced by:  funcestrcsetc  16721  fthestrcsetc  16722  fullestrcsetc  16723
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