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Theorem xpchom 17548
Description: Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
xpchomfval.t 𝑇 = (𝐶 ×c 𝐷)
xpchomfval.y 𝐵 = (Base‘𝑇)
xpchomfval.h 𝐻 = (Hom ‘𝐶)
xpchomfval.j 𝐽 = (Hom ‘𝐷)
xpchomfval.k 𝐾 = (Hom ‘𝑇)
xpchom.x (𝜑𝑋𝐵)
xpchom.y (𝜑𝑌𝐵)
Assertion
Ref Expression
xpchom (𝜑 → (𝑋𝐾𝑌) = (((1st𝑋)𝐻(1st𝑌)) × ((2nd𝑋)𝐽(2nd𝑌))))

Proof of Theorem xpchom
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpchom.x . 2 (𝜑𝑋𝐵)
2 xpchom.y . 2 (𝜑𝑌𝐵)
3 simpl 486 . . . . . 6 ((𝑢 = 𝑋𝑣 = 𝑌) → 𝑢 = 𝑋)
43fveq2d 6680 . . . . 5 ((𝑢 = 𝑋𝑣 = 𝑌) → (1st𝑢) = (1st𝑋))
5 simpr 488 . . . . . 6 ((𝑢 = 𝑋𝑣 = 𝑌) → 𝑣 = 𝑌)
65fveq2d 6680 . . . . 5 ((𝑢 = 𝑋𝑣 = 𝑌) → (1st𝑣) = (1st𝑌))
74, 6oveq12d 7190 . . . 4 ((𝑢 = 𝑋𝑣 = 𝑌) → ((1st𝑢)𝐻(1st𝑣)) = ((1st𝑋)𝐻(1st𝑌)))
83fveq2d 6680 . . . . 5 ((𝑢 = 𝑋𝑣 = 𝑌) → (2nd𝑢) = (2nd𝑋))
95fveq2d 6680 . . . . 5 ((𝑢 = 𝑋𝑣 = 𝑌) → (2nd𝑣) = (2nd𝑌))
108, 9oveq12d 7190 . . . 4 ((𝑢 = 𝑋𝑣 = 𝑌) → ((2nd𝑢)𝐽(2nd𝑣)) = ((2nd𝑋)𝐽(2nd𝑌)))
117, 10xpeq12d 5556 . . 3 ((𝑢 = 𝑋𝑣 = 𝑌) → (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))) = (((1st𝑋)𝐻(1st𝑌)) × ((2nd𝑋)𝐽(2nd𝑌))))
12 xpchomfval.t . . . 4 𝑇 = (𝐶 ×c 𝐷)
13 xpchomfval.y . . . 4 𝐵 = (Base‘𝑇)
14 xpchomfval.h . . . 4 𝐻 = (Hom ‘𝐶)
15 xpchomfval.j . . . 4 𝐽 = (Hom ‘𝐷)
16 xpchomfval.k . . . 4 𝐾 = (Hom ‘𝑇)
1712, 13, 14, 15, 16xpchomfval 17547 . . 3 𝐾 = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))
18 ovex 7205 . . . 4 ((1st𝑋)𝐻(1st𝑌)) ∈ V
19 ovex 7205 . . . 4 ((2nd𝑋)𝐽(2nd𝑌)) ∈ V
2018, 19xpex 7496 . . 3 (((1st𝑋)𝐻(1st𝑌)) × ((2nd𝑋)𝐽(2nd𝑌))) ∈ V
2111, 17, 20ovmpoa 7322 . 2 ((𝑋𝐵𝑌𝐵) → (𝑋𝐾𝑌) = (((1st𝑋)𝐻(1st𝑌)) × ((2nd𝑋)𝐽(2nd𝑌))))
221, 2, 21syl2anc 587 1 (𝜑 → (𝑋𝐾𝑌) = (((1st𝑋)𝐻(1st𝑌)) × ((2nd𝑋)𝐽(2nd𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114   × cxp 5523  cfv 6339  (class class class)co 7172  1st c1st 7714  2nd c2nd 7715  Basecbs 16588  Hom chom 16681   ×c cxpc 17536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7481  ax-cnex 10673  ax-resscn 10674  ax-1cn 10675  ax-icn 10676  ax-addcl 10677  ax-addrcl 10678  ax-mulcl 10679  ax-mulrcl 10680  ax-mulcom 10681  ax-addass 10682  ax-mulass 10683  ax-distr 10684  ax-i2m1 10685  ax-1ne0 10686  ax-1rid 10687  ax-rnegex 10688  ax-rrecex 10689  ax-cnre 10690  ax-pre-lttri 10691  ax-pre-lttrn 10692  ax-pre-ltadd 10693  ax-pre-mulgt0 10694
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7129  df-ov 7175  df-oprab 7176  df-mpo 7177  df-om 7602  df-1st 7716  df-2nd 7717  df-wrecs 7978  df-recs 8039  df-rdg 8077  df-1o 8133  df-er 8322  df-en 8558  df-dom 8559  df-sdom 8560  df-fin 8561  df-pnf 10757  df-mnf 10758  df-xr 10759  df-ltxr 10760  df-le 10761  df-sub 10952  df-neg 10953  df-nn 11719  df-2 11781  df-3 11782  df-4 11783  df-5 11784  df-6 11785  df-7 11786  df-8 11787  df-9 11788  df-n0 11979  df-z 12065  df-dec 12182  df-uz 12327  df-fz 12984  df-struct 16590  df-ndx 16591  df-slot 16592  df-base 16594  df-hom 16694  df-cco 16695  df-xpc 17540
This theorem is referenced by:  xpchom2  17554  xpccatid  17556  1stfcl  17565  2ndfcl  17566  xpcpropd  17576  evlfcl  17590  curf1cl  17596  hofcl  17627  yonedalem3  17648
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