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Theorem xpchom 16643
 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 472 . . . . . 6 ((𝑢 = 𝑋𝑣 = 𝑌) → 𝑢 = 𝑋)
43fveq2d 6107 . . . . 5 ((𝑢 = 𝑋𝑣 = 𝑌) → (1st𝑢) = (1st𝑋))
5 simpr 476 . . . . . 6 ((𝑢 = 𝑋𝑣 = 𝑌) → 𝑣 = 𝑌)
65fveq2d 6107 . . . . 5 ((𝑢 = 𝑋𝑣 = 𝑌) → (1st𝑣) = (1st𝑌))
74, 6oveq12d 6567 . . . 4 ((𝑢 = 𝑋𝑣 = 𝑌) → ((1st𝑢)𝐻(1st𝑣)) = ((1st𝑋)𝐻(1st𝑌)))
83fveq2d 6107 . . . . 5 ((𝑢 = 𝑋𝑣 = 𝑌) → (2nd𝑢) = (2nd𝑋))
95fveq2d 6107 . . . . 5 ((𝑢 = 𝑋𝑣 = 𝑌) → (2nd𝑣) = (2nd𝑌))
108, 9oveq12d 6567 . . . 4 ((𝑢 = 𝑋𝑣 = 𝑌) → ((2nd𝑢)𝐽(2nd𝑣)) = ((2nd𝑋)𝐽(2nd𝑌)))
117, 10xpeq12d 5064 . . 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 16642 . . 3 𝐾 = (𝑢𝐵, 𝑣𝐵 ↦ (((1st𝑢)𝐻(1st𝑣)) × ((2nd𝑢)𝐽(2nd𝑣))))
18 ovex 6577 . . . 4 ((1st𝑋)𝐻(1st𝑌)) ∈ V
19 ovex 6577 . . . 4 ((2nd𝑋)𝐽(2nd𝑌)) ∈ V
2018, 19xpex 6860 . . 3 (((1st𝑋)𝐻(1st𝑌)) × ((2nd𝑋)𝐽(2nd𝑌))) ∈ V
2111, 17, 20ovmpt2a 6689 . 2 ((𝑋𝐵𝑌𝐵) → (𝑋𝐾𝑌) = (((1st𝑋)𝐻(1st𝑌)) × ((2nd𝑋)𝐽(2nd𝑌))))
221, 2, 21syl2anc 691 1 (𝜑 → (𝑋𝐾𝑌) = (((1st𝑋)𝐻(1st𝑌)) × ((2nd𝑋)𝐽(2nd𝑌))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   × cxp 5036  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Basecbs 15695  Hom chom 15779   ×c cxpc 16631 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-hom 15793  df-cco 15794  df-xpc 16635 This theorem is referenced by:  xpchom2  16649  xpccatid  16651  1stfcl  16660  2ndfcl  16661  xpcpropd  16671  evlfcl  16685  curf1cl  16691  hofcl  16722  yonedalem3  16743
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