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Theorem xpchom 16736
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 473 . . . . . 6 ((𝑢 = 𝑋𝑣 = 𝑌) → 𝑢 = 𝑋)
43fveq2d 6154 . . . . 5 ((𝑢 = 𝑋𝑣 = 𝑌) → (1st𝑢) = (1st𝑋))
5 simpr 477 . . . . . 6 ((𝑢 = 𝑋𝑣 = 𝑌) → 𝑣 = 𝑌)
65fveq2d 6154 . . . . 5 ((𝑢 = 𝑋𝑣 = 𝑌) → (1st𝑣) = (1st𝑌))
74, 6oveq12d 6623 . . . 4 ((𝑢 = 𝑋𝑣 = 𝑌) → ((1st𝑢)𝐻(1st𝑣)) = ((1st𝑋)𝐻(1st𝑌)))
83fveq2d 6154 . . . . 5 ((𝑢 = 𝑋𝑣 = 𝑌) → (2nd𝑢) = (2nd𝑋))
95fveq2d 6154 . . . . 5 ((𝑢 = 𝑋𝑣 = 𝑌) → (2nd𝑣) = (2nd𝑌))
108, 9oveq12d 6623 . . . 4 ((𝑢 = 𝑋𝑣 = 𝑌) → ((2nd𝑢)𝐽(2nd𝑣)) = ((2nd𝑋)𝐽(2nd𝑌)))
117, 10xpeq12d 5105 . . 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 16735 . . 3 𝐾 = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))
18 ovex 6633 . . . 4 ((1st𝑋)𝐻(1st𝑌)) ∈ V
19 ovex 6633 . . . 4 ((2nd𝑋)𝐽(2nd𝑌)) ∈ V
2018, 19xpex 6916 . . 3 (((1st𝑋)𝐻(1st𝑌)) × ((2nd𝑋)𝐽(2nd𝑌))) ∈ V
2111, 17, 20ovmpt2a 6745 . 2 ((𝑋𝐵𝑌𝐵) → (𝑋𝐾𝑌) = (((1st𝑋)𝐻(1st𝑌)) × ((2nd𝑋)𝐽(2nd𝑌))))
221, 2, 21syl2anc 692 1 (𝜑 → (𝑋𝐾𝑌) = (((1st𝑋)𝐻(1st𝑌)) × ((2nd𝑋)𝐽(2nd𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992   × cxp 5077  cfv 5850  (class class class)co 6605  1st c1st 7114  2nd c2nd 7115  Basecbs 15776  Hom chom 15868   ×c cxpc 16724
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 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
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 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-3 11025  df-4 11026  df-5 11027  df-6 11028  df-7 11029  df-8 11030  df-9 11031  df-n0 11238  df-z 11323  df-dec 11438  df-uz 11632  df-fz 12266  df-struct 15778  df-ndx 15779  df-slot 15780  df-base 15781  df-hom 15882  df-cco 15883  df-xpc 16728
This theorem is referenced by:  xpchom2  16742  xpccatid  16744  1stfcl  16753  2ndfcl  16754  xpcpropd  16764  evlfcl  16778  curf1cl  16784  hofcl  16815  yonedalem3  16836
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