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Theorem funcestrcsetclem8 16610
 Description: Lemma 8 for funcestrcsetc 16612. (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 6086 . . . 4 ( I ↾ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))):((Base‘𝑌) ↑𝑚 (Base‘𝑋))–1-1-onto→((Base‘𝑌) ↑𝑚 (Base‘𝑋))
2 f1of 6050 . . . 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 7765 . . . . 5 (𝑓 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) → 𝑓:(Base‘𝑋)⟶(Base‘𝑌))
5 fvex 6113 . . . . . . . . . 10 (Base‘𝑌) ∈ V
6 fvex 6113 . . . . . . . . . 10 (Base‘𝑋) ∈ V
75, 6pm3.2i 470 . . . . . . . . 9 ((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V)
8 elmapg 7757 . . . . . . . . . 10 (((Base‘𝑌) ∈ V ∧ (Base‘𝑋) ∈ V) → (𝑓 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ↔ 𝑓:(Base‘𝑋)⟶(Base‘𝑌)))
98bicomd 212 . . . . . . . . 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 16603 . . . . . . . . . 10 ((𝜑𝑌𝐵) → (𝐹𝑌) = (Base‘𝑌))
1918adantrl 748 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑌) = (Base‘𝑌))
2012, 13, 14, 15, 16, 17funcestrcsetclem1 16603 . . . . . . . . . 10 ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
2120adantrr 749 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑋) = (Base‘𝑋))
2219, 21oveq12d 6567 . . . . . . . 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 2691 . . . . . 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 33 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑓 ∈ ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) → 𝑓 ∈ ((𝐹𝑌) ↑𝑚 (𝐹𝑋))))
2726ssrdv 3574 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((Base‘𝑌) ↑𝑚 (Base‘𝑋)) ⊆ ((𝐹𝑌) ↑𝑚 (𝐹𝑋)))
283, 27fssd 5970 . 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 2610 . . . 4 (Base‘𝑋) = (Base‘𝑋)
31 eqid 2610 . . . 4 (Base‘𝑌) = (Base‘𝑌)
3212, 13, 14, 15, 16, 17, 29, 30, 31funcestrcsetclem5 16607 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌) = ( I ↾ ((Base‘𝑌) ↑𝑚 (Base‘𝑋))))
3316adantr 480 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑈 ∈ WUni)
34 eqid 2610 . . . 4 (Hom ‘𝐸) = (Hom ‘𝐸)
3512, 16estrcbas 16588 . . . . . . . . 9 (𝜑𝑈 = (Base‘𝐸))
3635, 14syl6reqr 2663 . . . . . . . 8 (𝜑𝐵 = 𝑈)
3736eleq2d 2673 . . . . . . 7 (𝜑 → (𝑋𝐵𝑋𝑈))
3837biimpcd 238 . . . . . 6 (𝑋𝐵 → (𝜑𝑋𝑈))
3938adantr 480 . . . . 5 ((𝑋𝐵𝑌𝐵) → (𝜑𝑋𝑈))
4039impcom 445 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑋𝑈)
4136eleq2d 2673 . . . . . . 7 (𝜑 → (𝑌𝐵𝑌𝑈))
4241biimpd 218 . . . . . 6 (𝜑 → (𝑌𝐵𝑌𝑈))
4342adantld 482 . . . . 5 (𝜑 → ((𝑋𝐵𝑌𝐵) → 𝑌𝑈))
4443imp 444 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → 𝑌𝑈)
4512, 33, 34, 40, 44, 30, 31estrchom 16590 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋(Hom ‘𝐸)𝑌) = ((Base‘𝑌) ↑𝑚 (Base‘𝑋)))
46 eqid 2610 . . . 4 (Hom ‘𝑆) = (Hom ‘𝑆)
4712, 13, 14, 15, 16, 17funcestrcsetclem2 16604 . . . . 5 ((𝜑𝑋𝐵) → (𝐹𝑋) ∈ 𝑈)
4847adantrr 749 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑋) ∈ 𝑈)
4912, 13, 14, 15, 16, 17funcestrcsetclem2 16604 . . . . 5 ((𝜑𝑌𝐵) → (𝐹𝑌) ∈ 𝑈)
5049adantrl 748 . . . 4 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝐹𝑌) ∈ 𝑈)
5113, 33, 46, 48, 50setchom 16553 . . 3 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → ((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)) = ((𝐹𝑌) ↑𝑚 (𝐹𝑋)))
5232, 45, 51feq123d 5947 . 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 246 1 ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝐸)𝑌)⟶((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ↦ cmpt 4643   I cid 4948   ↾ cres 5040  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   ↑𝑚 cmap 7744  WUnicwun 9401  Basecbs 15695  Hom chom 15779  SetCatcsetc 16548  ExtStrCatcestrc 16585 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-wun 9403  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-hom 15793  df-cco 15794  df-setc 16549  df-estrc 16586 This theorem is referenced by:  funcestrcsetc  16612  fthestrcsetc  16613  fullestrcsetc  16614
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