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Theorem xpcco1st 17031
 Description: Value of composition in the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
xpcco1st.t 𝑇 = (𝐶 ×c 𝐷)
xpcco1st.b 𝐵 = (Base‘𝑇)
xpcco1st.k 𝐾 = (Hom ‘𝑇)
xpcco1st.o 𝑂 = (comp‘𝑇)
xpcco1st.x (𝜑𝑋𝐵)
xpcco1st.y (𝜑𝑌𝐵)
xpcco1st.z (𝜑𝑍𝐵)
xpcco1st.f (𝜑𝐹 ∈ (𝑋𝐾𝑌))
xpcco1st.g (𝜑𝐺 ∈ (𝑌𝐾𝑍))
xpcco1st.1 · = (comp‘𝐶)
Assertion
Ref Expression
xpcco1st (𝜑 → (1st ‘(𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹)) = ((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)))

Proof of Theorem xpcco1st
StepHypRef Expression
1 xpcco1st.t . . 3 𝑇 = (𝐶 ×c 𝐷)
2 xpcco1st.b . . 3 𝐵 = (Base‘𝑇)
3 xpcco1st.k . . 3 𝐾 = (Hom ‘𝑇)
4 xpcco1st.1 . . 3 · = (comp‘𝐶)
5 eqid 2770 . . 3 (comp‘𝐷) = (comp‘𝐷)
6 xpcco1st.o . . 3 𝑂 = (comp‘𝑇)
7 xpcco1st.x . . 3 (𝜑𝑋𝐵)
8 xpcco1st.y . . 3 (𝜑𝑌𝐵)
9 xpcco1st.z . . 3 (𝜑𝑍𝐵)
10 xpcco1st.f . . 3 (𝜑𝐹 ∈ (𝑋𝐾𝑌))
11 xpcco1st.g . . 3 (𝜑𝐺 ∈ (𝑌𝐾𝑍))
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11xpcco 17030 . 2 (𝜑 → (𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = ⟨((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)), ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩(comp‘𝐷)(2nd𝑍))(2nd𝐹))⟩)
13 ovex 6822 . . 3 ((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)) ∈ V
14 ovex 6822 . . 3 ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩(comp‘𝐷)(2nd𝑍))(2nd𝐹)) ∈ V
1513, 14op1std 7324 . 2 ((𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹) = ⟨((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)), ((2nd𝐺)(⟨(2nd𝑋), (2nd𝑌)⟩(comp‘𝐷)(2nd𝑍))(2nd𝐹))⟩ → (1st ‘(𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹)) = ((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)))
1612, 15syl 17 1 (𝜑 → (1st ‘(𝐺(⟨𝑋, 𝑌𝑂𝑍)𝐹)) = ((1st𝐺)(⟨(1st𝑋), (1st𝑌)⟩ · (1st𝑍))(1st𝐹)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1630   ∈ wcel 2144  ⟨cop 4320  ‘cfv 6031  (class class class)co 6792  1st c1st 7312  2nd c2nd 7313  Basecbs 16063  Hom chom 16159  compcco 16160   ×c cxpc 17015 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-3 11281  df-4 11282  df-5 11283  df-6 11284  df-7 11285  df-8 11286  df-9 11287  df-n0 11494  df-z 11579  df-dec 11695  df-uz 11888  df-fz 12533  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-hom 16173  df-cco 16174  df-xpc 17019 This theorem is referenced by:  1stfcl  17044
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