Theorem xpchom 18226

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 6909	. . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (1st ‘𝑢) = (1st ‘𝑋))
5		simpr 484	. . . . . 6 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → 𝑣 = 𝑌)
6	5	fveq2d 6909	. . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (1st ‘𝑣) = (1st ‘𝑌))
7	4, 6	oveq12d 7450	. . . 4 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → ((1st ‘𝑢)𝐻(1st ‘𝑣)) = ((1st ‘𝑋)𝐻(1st ‘𝑌)))
8	3	fveq2d 6909	. . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (2nd ‘𝑢) = (2nd ‘𝑋))
9	5	fveq2d 6909	. . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (2nd ‘𝑣) = (2nd ‘𝑌))
10	8, 9	oveq12d 7450	. . . 4 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → ((2nd ‘𝑢)𝐽(2nd ‘𝑣)) = ((2nd ‘𝑋)𝐽(2nd ‘𝑌)))
11	7, 10	xpeq12d 5715	. . 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 18225	. . 3 ⊢ 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1st ‘𝑢)𝐻(1st ‘𝑣)) × ((2nd ‘𝑢)𝐽(2nd ‘𝑣))))
18		ovex 7465	. . . 4 ⊢ ((1st ‘𝑋)𝐻(1st ‘𝑌)) ∈ V
19		ovex 7465	. . . 4 ⊢ ((2nd ‘𝑋)𝐽(2nd ‘𝑌)) ∈ V
20	18, 19	xpex 7774	. . 3 ⊢ (((1st ‘𝑋)𝐻(1st ‘𝑌)) × ((2nd ‘𝑋)𝐽(2nd ‘𝑌))) ∈ V
21	11, 17, 20	ovmpoa 7589	. 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 1539   ∈ wcel 2107   × cxp 5682  ‘cfv 6560  (class class class)co 7432  1^st c1st 8013  2^nd c2nd 8014  Basecbs 17248  Hom chom 17309   ×_c cxpc 18214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-struct 17185  df-slot 17220  df-ndx 17232  df-base 17249  df-hom 17322  df-cco 17323  df-xpc 18218

This theorem is referenced by:  xpchom2  18232  xpccatid  18234  1stfcl  18243  2ndfcl  18244  xpcpropd  18254  evlfcl  18268  curf1cl  18274  hofcl  18305  yonedalem3  18326  xpcfuchom  48978  dfswapf2  48985  swapf2a  48995  swapf2f1oaALT  49002  fucofulem2  49029