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Theorem 2ndf2 16776
 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 16774 . . 3 (𝜑𝑄 = ⟨(2nd𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)
8 fo2nd 7149 . . . . . 6 2nd :V–onto→V
9 fofun 6083 . . . . . 6 (2nd :V–onto→V → Fun 2nd )
108, 9ax-mp 5 . . . . 5 Fun 2nd
11 fvex 6168 . . . . . 6 (Base‘𝑇) ∈ V
122, 11eqeltri 2694 . . . . 5 𝐵 ∈ V
13 resfunexg 6444 . . . . 5 ((Fun 2nd𝐵 ∈ V) → (2nd𝐵) ∈ V)
1410, 12, 13mp2an 707 . . . 4 (2nd𝐵) ∈ V
1512, 12mpt2ex 7207 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))) ∈ V
1614, 15op2ndd 7139 . . 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 6633 . . 3 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → (𝑥𝐻𝑦) = (𝑅𝐻𝑆))
2120reseq2d 5366 . 2 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → (2nd ↾ (𝑥𝐻𝑦)) = (2nd ↾ (𝑅𝐻𝑆)))
22 2ndf1.p . 2 (𝜑𝑅𝐵)
23 2ndf2.p . 2 (𝜑𝑆𝐵)
24 ovex 6643 . . . 4 (𝑅𝐻𝑆) ∈ V
25 resfunexg 6444 . . . 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 6753 1 (𝜑 → (𝑅(2nd𝑄)𝑆) = (2nd ↾ (𝑅𝐻𝑆)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  Vcvv 3190  ⟨cop 4161   ↾ cres 5086  Fun wfun 5851  –onto→wfo 5855  ‘cfv 5857  (class class class)co 6615   ↦ cmpt2 6617  2nd c2nd 7127  Basecbs 15800  Hom chom 15892  Catccat 16265   ×c cxpc 16748   2ndF c2ndf 16750 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 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  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 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-5 11042  df-6 11043  df-7 11044  df-8 11045  df-9 11046  df-n0 11253  df-z 11338  df-dec 11454  df-uz 11648  df-fz 12285  df-struct 15802  df-ndx 15803  df-slot 15804  df-base 15805  df-hom 15906  df-cco 15907  df-xpc 16752  df-2ndf 16754 This theorem is referenced by:  2ndfcl  16778  prf2nd  16785  1st2ndprf  16786  uncf2  16817  curf2ndf  16827  yonedalem22  16858
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