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Theorem 2ndf2 18156
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 18154 . . 3 (𝜑𝑄 = ⟨(2nd𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩)
8 fo2nd 7990 . . . . . 6 2nd :V–onto→V
9 fofun 6797 . . . . . 6 (2nd :V–onto→V → Fun 2nd )
108, 9ax-mp 5 . . . . 5 Fun 2nd
112fvexi 6896 . . . . 5 𝐵 ∈ V
12 resfunexg 7209 . . . . 5 ((Fun 2nd𝐵 ∈ V) → (2nd𝐵) ∈ V)
1310, 11, 12mp2an 689 . . . 4 (2nd𝐵) ∈ V
1411, 11mpoex 8060 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))) ∈ V
1513, 14op2ndd 7980 . . 3 (𝑄 = ⟨(2nd𝐵), (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦)))⟩ → (2nd𝑄) = (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))))
167, 15syl 17 . 2 (𝜑 → (2nd𝑄) = (𝑥𝐵, 𝑦𝐵 ↦ (2nd ↾ (𝑥𝐻𝑦))))
17 simprl 768 . . . 4 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → 𝑥 = 𝑅)
18 simprr 770 . . . 4 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → 𝑦 = 𝑆)
1917, 18oveq12d 7420 . . 3 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → (𝑥𝐻𝑦) = (𝑅𝐻𝑆))
2019reseq2d 5972 . 2 ((𝜑 ∧ (𝑥 = 𝑅𝑦 = 𝑆)) → (2nd ↾ (𝑥𝐻𝑦)) = (2nd ↾ (𝑅𝐻𝑆)))
21 2ndf1.p . 2 (𝜑𝑅𝐵)
22 2ndf2.p . 2 (𝜑𝑆𝐵)
23 ovex 7435 . . . 4 (𝑅𝐻𝑆) ∈ V
24 resfunexg 7209 . . . 4 ((Fun 2nd ∧ (𝑅𝐻𝑆) ∈ V) → (2nd ↾ (𝑅𝐻𝑆)) ∈ V)
2510, 23, 24mp2an 689 . . 3 (2nd ↾ (𝑅𝐻𝑆)) ∈ V
2625a1i 11 . 2 (𝜑 → (2nd ↾ (𝑅𝐻𝑆)) ∈ V)
2716, 20, 21, 22, 26ovmpod 7553 1 (𝜑 → (𝑅(2nd𝑄)𝑆) = (2nd ↾ (𝑅𝐻𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  Vcvv 3466  cop 4627  cres 5669  Fun wfun 6528  ontowfo 6532  cfv 6534  (class class class)co 7402  cmpo 7404  2nd c2nd 7968  Basecbs 17149  Hom chom 17213  Catccat 17613   ×c cxpc 18128   2ndF c2ndf 18130
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-tp 4626  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12472  df-z 12558  df-dec 12677  df-slot 17120  df-ndx 17132  df-base 17150  df-hom 17226  df-cco 17227  df-xpc 18132  df-2ndf 18134
This theorem is referenced by:  2ndfcl  18158  prf2nd  18165  1st2ndprf  18166  uncf2  18198  curf2ndf  18208  yonedalem22  18239
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