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Theorem xpchom 18176
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 481 . . . . . 6 ((𝑢 = 𝑋𝑣 = 𝑌) → 𝑢 = 𝑋)
43fveq2d 6904 . . . . 5 ((𝑢 = 𝑋𝑣 = 𝑌) → (1st𝑢) = (1st𝑋))
5 simpr 483 . . . . . 6 ((𝑢 = 𝑋𝑣 = 𝑌) → 𝑣 = 𝑌)
65fveq2d 6904 . . . . 5 ((𝑢 = 𝑋𝑣 = 𝑌) → (1st𝑣) = (1st𝑌))
74, 6oveq12d 7442 . . . 4 ((𝑢 = 𝑋𝑣 = 𝑌) → ((1st𝑢)𝐻(1st𝑣)) = ((1st𝑋)𝐻(1st𝑌)))
83fveq2d 6904 . . . . 5 ((𝑢 = 𝑋𝑣 = 𝑌) → (2nd𝑢) = (2nd𝑋))
95fveq2d 6904 . . . . 5 ((𝑢 = 𝑋𝑣 = 𝑌) → (2nd𝑣) = (2nd𝑌))
108, 9oveq12d 7442 . . . 4 ((𝑢 = 𝑋𝑣 = 𝑌) → ((2nd𝑢)𝐽(2nd𝑣)) = ((2nd𝑋)𝐽(2nd𝑌)))
117, 10xpeq12d 5711 . . 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 18175 . . 3 𝐾 = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))
18 ovex 7457 . . . 4 ((1st𝑋)𝐻(1st𝑌)) ∈ V
19 ovex 7457 . . . 4 ((2nd𝑋)𝐽(2nd𝑌)) ∈ V
2018, 19xpex 7759 . . 3 (((1st𝑋)𝐻(1st𝑌)) × ((2nd𝑋)𝐽(2nd𝑌))) ∈ V
2111, 17, 20ovmpoa 7580 . 2 ((𝑋𝐵𝑌𝐵) → (𝑋𝐾𝑌) = (((1st𝑋)𝐻(1st𝑌)) × ((2nd𝑋)𝐽(2nd𝑌))))
221, 2, 21syl2anc 582 1 (𝜑 → (𝑋𝐾𝑌) = (((1st𝑋)𝐻(1st𝑌)) × ((2nd𝑋)𝐽(2nd𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098   × cxp 5678  cfv 6551  (class class class)co 7424  1st c1st 7995  2nd c2nd 7996  Basecbs 17185  Hom chom 17249   ×c cxpc 18164
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 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744  ax-cnex 11200  ax-resscn 11201  ax-1cn 11202  ax-icn 11203  ax-addcl 11204  ax-addrcl 11205  ax-mulcl 11206  ax-mulrcl 11207  ax-mulcom 11208  ax-addass 11209  ax-mulass 11210  ax-distr 11211  ax-i2m1 11212  ax-1ne0 11213  ax-1rid 11214  ax-rnegex 11215  ax-rrecex 11216  ax-cnre 11217  ax-pre-lttri 11218  ax-pre-lttrn 11219  ax-pre-ltadd 11220  ax-pre-mulgt0 11221
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7875  df-1st 7997  df-2nd 7998  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-rdg 8435  df-1o 8491  df-er 8729  df-en 8969  df-dom 8970  df-sdom 8971  df-fin 8972  df-pnf 11286  df-mnf 11287  df-xr 11288  df-ltxr 11289  df-le 11290  df-sub 11482  df-neg 11483  df-nn 12249  df-2 12311  df-3 12312  df-4 12313  df-5 12314  df-6 12315  df-7 12316  df-8 12317  df-9 12318  df-n0 12509  df-z 12595  df-dec 12714  df-uz 12859  df-fz 13523  df-struct 17121  df-slot 17156  df-ndx 17168  df-base 17186  df-hom 17262  df-cco 17263  df-xpc 18168
This theorem is referenced by:  xpchom2  18182  xpccatid  18184  1stfcl  18193  2ndfcl  18194  xpcpropd  18205  evlfcl  18219  curf1cl  18225  hofcl  18256  yonedalem3  18277
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