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Theorem 2ndf1 17437
Description: Value of the first projection on an object. (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)
Assertion
Ref Expression
2ndf1 (𝜑 → ((1st𝑄)‘𝑅) = (2nd𝑅))

Proof of Theorem 2ndf1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1stfval.t . . . . 5 𝑇 = (𝐶 ×c 𝐷)
2 1stfval.b . . . . 5 𝐵 = (Base‘𝑇)
3 1stfval.h . . . . 5 𝐻 = (Hom ‘𝑇)
4 1stfval.c . . . . 5 (𝜑𝐶 ∈ Cat)
5 1stfval.d . . . . 5 (𝜑𝐷 ∈ Cat)
6 2ndfval.p . . . . 5 𝑄 = (𝐶 2ndF 𝐷)
71, 2, 3, 4, 5, 62ndfval 17436 . . . 4 (𝜑𝑄 = ⟨(2nd𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)
8 fo2nd 7702 . . . . . . 7 2nd :V–onto→V
9 fofun 6584 . . . . . . 7 (2nd :V–onto→V → Fun 2nd )
108, 9ax-mp 5 . . . . . 6 Fun 2nd
112fvexi 6677 . . . . . 6 𝐵 ∈ V
12 resfunexg 6970 . . . . . 6 ((Fun 2nd𝐵 ∈ V) → (2nd𝐵) ∈ V)
1310, 11, 12mp2an 690 . . . . 5 (2nd𝐵) ∈ V
1411, 11mpoex 7769 . . . . 5 (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))) ∈ V
1513, 14op1std 7691 . . . 4 (𝑄 = ⟨(2nd𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩ → (1st𝑄) = (2nd𝐵))
167, 15syl 17 . . 3 (𝜑 → (1st𝑄) = (2nd𝐵))
1716fveq1d 6665 . 2 (𝜑 → ((1st𝑄)‘𝑅) = ((2nd𝐵)‘𝑅))
18 2ndf1.p . . 3 (𝜑𝑅𝐵)
1918fvresd 6683 . 2 (𝜑 → ((2nd𝐵)‘𝑅) = (2nd𝑅))
2017, 19eqtrd 2854 1 (𝜑 → ((1st𝑄)‘𝑅) = (2nd𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  Vcvv 3493  cop 4565  cres 5550  Fun wfun 6342  ontowfo 6346  cfv 6348  (class class class)co 7148  cmpo 7150  1st c1st 7679  2nd c2nd 7680  Basecbs 16475  Hom chom 16568  Catccat 16927   ×c cxpc 17410   2ndF c2ndf 17412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-ndx 16478  df-slot 16479  df-base 16481  df-hom 16581  df-cco 16582  df-xpc 17414  df-2ndf 17416
This theorem is referenced by:  prf2nd  17447  1st2ndprf  17448  uncf1  17478  uncf2  17479  curf2ndf  17489  yonedalem21  17515  yonedalem22  17520
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