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Theorem 1stf2 17438
 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 17436 . . 3 (𝜑𝑃 = ⟨(1st𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩)
8 fo1st 7694 . . . . . 6 1st :V–onto→V
9 fofun 6567 . . . . . 6 (1st :V–onto→V → Fun 1st )
108, 9ax-mp 5 . . . . 5 Fun 1st
112fvexi 6660 . . . . 5 𝐵 ∈ V
12 resfunexg 6956 . . . . 5 ((Fun 1st𝐵 ∈ V) → (1st𝐵) ∈ V)
1310, 11, 12mp2an 691 . . . 4 (1st𝐵) ∈ V
1411, 11mpoex 7763 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦))) ∈ V
1513, 14op2ndd 7685 . . 3 (𝑃 = ⟨(1st𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦)))⟩ → (2nd𝑃) = (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦))))
167, 15syl 17 . 2 (𝜑 → (2nd𝑃) = (𝑥𝐵, 𝑦𝐵 ↦ (1st ↾ (𝑥𝐻𝑦))))
17 simprl 770 . . . 4 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → 𝑥 = 𝑅)
18 simprr 772 . . . 4 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → 𝑦 = 𝑆)
1917, 18oveq12d 7154 . . 3 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → (𝑥𝐻𝑦) = (𝑅𝐻𝑆))
2019reseq2d 5819 . 2 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → (1st ↾ (𝑥𝐻𝑦)) = (1st ↾ (𝑅𝐻𝑆)))
21 1stf1.p . 2 (𝜑𝑅𝐵)
22 1stf2.p . 2 (𝜑𝑆𝐵)
23 ovex 7169 . . . 4 (𝑅𝐻𝑆) ∈ V
24 resfunexg 6956 . . . 4 ((Fun 1st ∧ (𝑅𝐻𝑆) ∈ V) → (1st ↾ (𝑅𝐻𝑆)) ∈ V)
2510, 23, 24mp2an 691 . . 3 (1st ↾ (𝑅𝐻𝑆)) ∈ V
2625a1i 11 . 2 (𝜑 → (1st ↾ (𝑅𝐻𝑆)) ∈ V)
2716, 20, 21, 22, 26ovmpod 7283 1 (𝜑 → (𝑅(2nd𝑃)𝑆) = (1st ↾ (𝑅𝐻𝑆)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ⟨cop 4531   ↾ cres 5522  Fun wfun 6319  –onto→wfo 6323  ‘cfv 6325  (class class class)co 7136   ∈ cmpo 7138  1st c1st 7672  2nd c2nd 7673  Basecbs 16478  Hom chom 16571  Catccat 16930   ×c cxpc 17413   1stF c1stf 17414 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  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 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-er 8275  df-en 8496  df-dom 8497  df-sdom 8498  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11629  df-2 11691  df-3 11692  df-4 11693  df-5 11694  df-6 11695  df-7 11696  df-8 11697  df-9 11698  df-n0 11889  df-z 11973  df-dec 12090  df-ndx 16481  df-slot 16482  df-base 16484  df-hom 16584  df-cco 16585  df-xpc 17417  df-1stf 17418 This theorem is referenced by:  1stfcl  17442  prf1st  17449  1st2ndprf  17451  uncf2  17482  diag12  17489  diag2  17490  yonedalem22  17523
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