Theorem xpchom 18190

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	⊢ (𝜑 → 𝑋 ∈ 𝐵)
xpchom.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
xpchom	⊢ (𝜑 → (𝑋𝐾𝑌) = (((1st ‘𝑋)𝐻(1st ‘𝑌)) × ((2nd ‘𝑋)𝐽(2nd ‘𝑌))))

Proof of Theorem xpchom

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpchom.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
2		xpchom.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
3		simpl 482	. . . . . 6 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → 𝑢 = 𝑋)
4	3	fveq2d 6879	. . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (1st ‘𝑢) = (1st ‘𝑋))
5		simpr 484	. . . . . 6 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → 𝑣 = 𝑌)
6	5	fveq2d 6879	. . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (1st ‘𝑣) = (1st ‘𝑌))
7	4, 6	oveq12d 7421	. . . 4 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → ((1st ‘𝑢)𝐻(1st ‘𝑣)) = ((1st ‘𝑋)𝐻(1st ‘𝑌)))
8	3	fveq2d 6879	. . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (2nd ‘𝑢) = (2nd ‘𝑋))
9	5	fveq2d 6879	. . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (2nd ‘𝑣) = (2nd ‘𝑌))
10	8, 9	oveq12d 7421	. . . 4 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → ((2nd ‘𝑢)𝐽(2nd ‘𝑣)) = ((2nd ‘𝑋)𝐽(2nd ‘𝑌)))
11	7, 10	xpeq12d 5685	. . 3 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (((1st ‘𝑢)𝐻(1st ‘𝑣)) × ((2nd ‘𝑢)𝐽(2nd ‘𝑣))) = (((1st ‘𝑋)𝐻(1st ‘𝑌)) × ((2nd ‘𝑋)𝐽(2nd ‘𝑌))))
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)
17	12, 13, 14, 15, 16	xpchomfval 18189	. . 3 ⊢ 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1st ‘𝑢)𝐻(1st ‘𝑣)) × ((2nd ‘𝑢)𝐽(2nd ‘𝑣))))
18		ovex 7436	. . . 4 ⊢ ((1st ‘𝑋)𝐻(1st ‘𝑌)) ∈ V
19		ovex 7436	. . . 4 ⊢ ((2nd ‘𝑋)𝐽(2nd ‘𝑌)) ∈ V
20	18, 19	xpex 7745	. . 3 ⊢ (((1st ‘𝑋)𝐻(1st ‘𝑌)) × ((2nd ‘𝑋)𝐽(2nd ‘𝑌))) ∈ V
21	11, 17, 20	ovmpoa 7560	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐾𝑌) = (((1st ‘𝑋)𝐻(1st ‘𝑌)) × ((2nd ‘𝑋)𝐽(2nd ‘𝑌))))
22	1, 2, 21	syl2anc 584	1 ⊢ (𝜑 → (𝑋𝐾𝑌) = (((1st ‘𝑋)𝐻(1st ‘𝑌)) × ((2nd ‘𝑋)𝐽(2nd ‘𝑌))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108   × cxp 5652  ‘cfv 6530  (class class class)co 7403  1^st c1st 7984  2^nd c2nd 7985  Basecbs 17226  Hom chom 17280   ×_c cxpc 18178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-riota 7360  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7860  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8717  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11466  df-neg 11467  df-nn 12239  df-2 12301  df-3 12302  df-4 12303  df-5 12304  df-6 12305  df-7 12306  df-8 12307  df-9 12308  df-n0 12500  df-z 12587  df-dec 12707  df-uz 12851  df-fz 13523  df-struct 17164  df-slot 17199  df-ndx 17211  df-base 17227  df-hom 17293  df-cco 17294  df-xpc 18182

This theorem is referenced by:  xpchom2  18196  xpccatid  18198  1stfcl  18207  2ndfcl  18208  xpcpropd  18218  evlfcl  18232  curf1cl  18238  hofcl  18269  yonedalem3  18290  xpcfuchom  49119  dfswapf2  49126  swapf2a  49136  swapf2f1oaALT  49143