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Theorem 2ndf2 16817
Description: Value of the first projection on a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
1stfval.t 𝑇 = (𝐶 ×c 𝐷)
1stfval.b 𝐵 = (Base‘𝑇)
1stfval.h 𝐻 = (Hom ‘𝑇)
1stfval.c (𝜑𝐶 ∈ Cat)
1stfval.d (𝜑𝐷 ∈ Cat)
2ndfval.p 𝑄 = (𝐶 2ndF 𝐷)
2ndf1.p (𝜑𝑅𝐵)
2ndf2.p (𝜑𝑆𝐵)
Assertion
Ref Expression
2ndf2 (𝜑 → (𝑅(2nd𝑄)𝑆) = (2nd ↾ (𝑅𝐻𝑆)))

Proof of Theorem 2ndf2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1stfval.t . . . 4 𝑇 = (𝐶 ×c 𝐷)
2 1stfval.b . . . 4 𝐵 = (Base‘𝑇)
3 1stfval.h . . . 4 𝐻 = (Hom ‘𝑇)
4 1stfval.c . . . 4 (𝜑𝐶 ∈ Cat)
5 1stfval.d . . . 4 (𝜑𝐷 ∈ Cat)
6 2ndfval.p . . . 4 𝑄 = (𝐶 2ndF 𝐷)
71, 2, 3, 4, 5, 62ndfval 16815 . . 3 (𝜑𝑄 = ⟨(2nd𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)
8 fo2nd 7174 . . . . . 6 2nd :V–onto→V
9 fofun 6103 . . . . . 6 (2nd :V–onto→V → Fun 2nd )
108, 9ax-mp 5 . . . . 5 Fun 2nd
11 fvex 6188 . . . . . 6 (Base‘𝑇) ∈ V
122, 11eqeltri 2695 . . . . 5 𝐵 ∈ V
13 resfunexg 6464 . . . . 5 ((Fun 2nd𝐵 ∈ V) → (2nd𝐵) ∈ V)
1410, 12, 13mp2an 707 . . . 4 (2nd𝐵) ∈ V
1512, 12mpt2ex 7232 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))) ∈ V
1614, 15op2ndd 7164 . . 3 (𝑄 = ⟨(2nd𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩ → (2nd𝑄) = (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))))
177, 16syl 17 . 2 (𝜑 → (2nd𝑄) = (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))))
18 simprl 793 . . . 4 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → 𝑥 = 𝑅)
19 simprr 795 . . . 4 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → 𝑦 = 𝑆)
2018, 19oveq12d 6653 . . 3 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → (𝑥𝐻𝑦) = (𝑅𝐻𝑆))
2120reseq2d 5385 . 2 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → (2nd ↾ (𝑥𝐻𝑦)) = (2nd ↾ (𝑅𝐻𝑆)))
22 2ndf1.p . 2 (𝜑𝑅𝐵)
23 2ndf2.p . 2 (𝜑𝑆𝐵)
24 ovex 6663 . . . 4 (𝑅𝐻𝑆) ∈ V
25 resfunexg 6464 . . . 4 ((Fun 2nd ∧ (𝑅𝐻𝑆) ∈ V) → (2nd ↾ (𝑅𝐻𝑆)) ∈ V)
2610, 24, 25mp2an 707 . . 3 (2nd ↾ (𝑅𝐻𝑆)) ∈ V
2726a1i 11 . 2 (𝜑 → (2nd ↾ (𝑅𝐻𝑆)) ∈ V)
2817, 21, 22, 23, 27ovmpt2d 6773 1 (𝜑 → (𝑅(2nd𝑄)𝑆) = (2nd ↾ (𝑅𝐻𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  Vcvv 3195  cop 4174  cres 5106  Fun wfun 5870  ontowfo 5874  cfv 5876  (class class class)co 6635  cmpt2 6637  2nd c2nd 7152  Basecbs 15838  Hom chom 15933  Catccat 16306   ×c cxpc 16789   2ndF c2ndf 16791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-2 11064  df-3 11065  df-4 11066  df-5 11067  df-6 11068  df-7 11069  df-8 11070  df-9 11071  df-n0 11278  df-z 11363  df-dec 11479  df-uz 11673  df-fz 12312  df-struct 15840  df-ndx 15841  df-slot 15842  df-base 15844  df-hom 15947  df-cco 15948  df-xpc 16793  df-2ndf 16795
This theorem is referenced by:  2ndfcl  16819  prf2nd  16826  1st2ndprf  16827  uncf2  16858  curf2ndf  16868  yonedalem22  16899
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