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Theorem 1stf2 17291
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)
1stfval.p 𝑃 = (𝐶 1stF 𝐷)
1stf1.p (𝜑𝑅𝐵)
1stf2.p (𝜑𝑆𝐵)
Assertion
Ref Expression
1stf2 (𝜑 → (𝑅(2nd𝑃)𝑆) = (1st ↾ (𝑅𝐻𝑆)))

Proof of Theorem 1stf2
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 1stfval.p . . . 4 𝑃 = (𝐶 1stF 𝐷)
71, 2, 3, 4, 5, 61stfval 17289 . . 3 (𝜑𝑃 = ⟨(1st𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)
8 fo1st 7514 . . . . . 6 1st :V–onto→V
9 fofun 6414 . . . . . 6 (1st :V–onto→V → Fun 1st )
108, 9ax-mp 5 . . . . 5 Fun 1st
112fvexi 6507 . . . . 5 𝐵 ∈ V
12 resfunexg 6798 . . . . 5 ((Fun 1st𝐵 ∈ V) → (1st𝐵) ∈ V)
1310, 11, 12mp2an 679 . . . 4 (1st𝐵) ∈ V
1411, 11mpoex 7578 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦))) ∈ V
1513, 14op2ndd 7505 . . 3 (𝑃 = ⟨(1st𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩ → (2nd𝑃) = (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦))))
167, 15syl 17 . 2 (𝜑 → (2nd𝑃) = (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦))))
17 simprl 758 . . . 4 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → 𝑥 = 𝑅)
18 simprr 760 . . . 4 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → 𝑦 = 𝑆)
1917, 18oveq12d 6988 . . 3 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → (𝑥𝐻𝑦) = (𝑅𝐻𝑆))
2019reseq2d 5688 . 2 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → (1st ↾ (𝑥𝐻𝑦)) = (1st ↾ (𝑅𝐻𝑆)))
21 1stf1.p . 2 (𝜑𝑅𝐵)
22 1stf2.p . 2 (𝜑𝑆𝐵)
23 ovex 7002 . . . 4 (𝑅𝐻𝑆) ∈ V
24 resfunexg 6798 . . . 4 ((Fun 1st ∧ (𝑅𝐻𝑆) ∈ V) → (1st ↾ (𝑅𝐻𝑆)) ∈ V)
2510, 23, 24mp2an 679 . . 3 (1st ↾ (𝑅𝐻𝑆)) ∈ V
2625a1i 11 . 2 (𝜑 → (1st ↾ (𝑅𝐻𝑆)) ∈ V)
2716, 20, 21, 22, 26ovmpod 7112 1 (𝜑 → (𝑅(2nd𝑃)𝑆) = (1st ↾ (𝑅𝐻𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wcel 2048  Vcvv 3409  cop 4441  cres 5402  Fun wfun 6176  ontowfo 6180  cfv 6182  (class class class)co 6970  cmpo 6972  1st c1st 7492  2nd c2nd 7493  Basecbs 16329  Hom chom 16422  Catccat 16783   ×c cxpc 17266   1stF c1stf 17267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10383  ax-resscn 10384  ax-1cn 10385  ax-icn 10386  ax-addcl 10387  ax-addrcl 10388  ax-mulcl 10389  ax-mulrcl 10390  ax-mulcom 10391  ax-addass 10392  ax-mulass 10393  ax-distr 10394  ax-i2m1 10395  ax-1ne0 10396  ax-1rid 10397  ax-rnegex 10398  ax-rrecex 10399  ax-cnre 10400  ax-pre-lttri 10401  ax-pre-lttrn 10402  ax-pre-ltadd 10403  ax-pre-mulgt0 10404
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-1st 7494  df-2nd 7495  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-er 8081  df-en 8299  df-dom 8300  df-sdom 8301  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-sub 10664  df-neg 10665  df-nn 11432  df-2 11496  df-3 11497  df-4 11498  df-5 11499  df-6 11500  df-7 11501  df-8 11502  df-9 11503  df-n0 11701  df-z 11787  df-dec 11905  df-ndx 16332  df-slot 16333  df-base 16335  df-hom 16435  df-cco 16436  df-xpc 17270  df-1stf 17271
This theorem is referenced by:  1stfcl  17295  prf1st  17302  1st2ndprf  17304  uncf2  17335  diag12  17342  diag2  17343  yonedalem22  17376
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