xpcbas - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > xpcbas		Structured version Visualization version GIF version

Theorem xpcbas 17178

Description: Set of objects of the binary product of categories. (Contributed by Mario Carneiro, 10-Jan-2017.)

**Hypotheses**
Ref	Expression
xpcbas.t	⊢ 𝑇 = (𝐶 ×_c 𝐷)
xpcbas.x	⊢ 𝑋 = (Base‘𝐶)
xpcbas.y	⊢ 𝑌 = (Base‘𝐷)

**Assertion**
Ref	Expression
xpcbas	⊢ (𝑋 × 𝑌) = (Base‘𝑇)

Proof of Theorem xpcbas Dummy variables 𝑓 𝑔 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpcbas.t	. . . . 5 ⊢ 𝑇 = (𝐶 ×_c 𝐷)
2		xpcbas.x	. . . . 5 ⊢ 𝑋 = (Base‘𝐶)
3		xpcbas.y	. . . . 5 ⊢ 𝑌 = (Base‘𝐷)
4		eqid 2825	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
5		eqid 2825	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
6		eqid 2825	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
7		eqid 2825	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
8		simpl 476	. . . . 5 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐶 ∈ V)
9		simpr 479	. . . . 5 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐷 ∈ V)
10		eqidd 2826	. . . . 5 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = (𝑋 × 𝑌))
11		eqidd 2826	. . . . 5 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣)))) = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣)))))
12		eqidd 2826	. . . . 5 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑥 ∈ ((𝑋 × 𝑌) × (𝑋 × 𝑌)), 𝑦 ∈ (𝑋 × 𝑌) ↦ (𝑔 ∈ ((2^nd ‘𝑥)(𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))))𝑦), 𝑓 ∈ ((𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))))‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐶)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐷)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩)) = (𝑥 ∈ ((𝑋 × 𝑌) × (𝑋 × 𝑌)), 𝑦 ∈ (𝑋 × 𝑌) ↦ (𝑔 ∈ ((2^nd ‘𝑥)(𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))))𝑦), 𝑓 ∈ ((𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))))‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐶)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐷)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩)))
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	xpcval 17177	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑇 = {⟨(Base‘ndx), (𝑋 × 𝑌)⟩, ⟨(Hom ‘ndx), (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))))⟩, ⟨(comp‘ndx), (𝑥 ∈ ((𝑋 × 𝑌) × (𝑋 × 𝑌)), 𝑦 ∈ (𝑋 × 𝑌) ↦ (𝑔 ∈ ((2^nd ‘𝑥)(𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))))𝑦), 𝑓 ∈ ((𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))))‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐶)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐷)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩})
14		catstr 16976	. . . 4 ⊢ {⟨(Base‘ndx), (𝑋 × 𝑌)⟩, ⟨(Hom ‘ndx), (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))))⟩, ⟨(comp‘ndx), (𝑥 ∈ ((𝑋 × 𝑌) × (𝑋 × 𝑌)), 𝑦 ∈ (𝑋 × 𝑌) ↦ (𝑔 ∈ ((2^nd ‘𝑥)(𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))))𝑦), 𝑓 ∈ ((𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))))‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐶)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐷)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩} Struct ⟨1, ;15⟩
15		baseid 16289	. . . 4 ⊢ Base = Slot (Base‘ndx)
16		snsstp1 4567	. . . 4 ⊢ {⟨(Base‘ndx), (𝑋 × 𝑌)⟩} ⊆ {⟨(Base‘ndx), (𝑋 × 𝑌)⟩, ⟨(Hom ‘ndx), (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))))⟩, ⟨(comp‘ndx), (𝑥 ∈ ((𝑋 × 𝑌) × (𝑋 × 𝑌)), 𝑦 ∈ (𝑋 × 𝑌) ↦ (𝑔 ∈ ((2^nd ‘𝑥)(𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))))𝑦), 𝑓 ∈ ((𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))))‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐶)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐷)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩}
17	2	fvexi 6451	. . . . . 6 ⊢ 𝑋 ∈ V
18	3	fvexi 6451	. . . . . 6 ⊢ 𝑌 ∈ V
19	17, 18	xpex 7228	. . . . 5 ⊢ (𝑋 × 𝑌) ∈ V
20	19	a1i 11	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) ∈ V)
21		eqid 2825	. . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇)
22	13, 14, 15, 16, 20, 21	strfv3 16278	. . 3 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (Base‘𝑇) = (𝑋 × 𝑌))
23	22	eqcomd 2831	. 2 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = (Base‘𝑇))
24		base0 16282	. . 3 ⊢ ∅ = (Base‘∅)
25		fvprc 6430	. . . . . 6 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = ∅)
26	2, 25	syl5eq 2873	. . . . 5 ⊢ (¬ 𝐶 ∈ V → 𝑋 = ∅)
27		fvprc 6430	. . . . . 6 ⊢ (¬ 𝐷 ∈ V → (Base‘𝐷) = ∅)
28	3, 27	syl5eq 2873	. . . . 5 ⊢ (¬ 𝐷 ∈ V → 𝑌 = ∅)
29	26, 28	orim12i 937	. . . 4 ⊢ ((¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V) → (𝑋 = ∅ ∨ 𝑌 = ∅))
30		ianor 1009	. . . 4 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V))
31		xpeq0 5799	. . . 4 ⊢ ((𝑋 × 𝑌) = ∅ ↔ (𝑋 = ∅ ∨ 𝑌 = ∅))
32	29, 30, 31	3imtr4i 284	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = ∅)
33		fnxpc 17176	. . . . . . 7 ⊢ ×_c Fn (V × V)
34		fndm 6227	. . . . . . 7 ⊢ ( ×_c Fn (V × V) → dom ×_c = (V × V))
35	33, 34	ax-mp 5	. . . . . 6 ⊢ dom ×_c = (V × V)
36	35	ndmov 7083	. . . . 5 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 ×_c 𝐷) = ∅)
37	1, 36	syl5eq 2873	. . . 4 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑇 = ∅)
38	37	fveq2d 6441	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (Base‘𝑇) = (Base‘∅))
39	24, 32, 38	3eqtr4a 2887	. 2 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = (Base‘𝑇))
40	23, 39	pm2.61i 177	1 ⊢ (𝑋 × 𝑌) = (Base‘𝑇)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 386 ∨ wo 878 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ∅c0 4146 {ctp 4403 ⟨cop 4405 × cxp 5344 dom cdm 5346 Fn wfn 6122 ‘cfv 6127 (class class class)co 6910 ↦ cmpt2 6912 1^st c1st 7431 2^nd c2nd 7432 1c1 10260 5c5 11416 cdc 11828 ndxcnx 16226 Basecbs 16229 Hom chom 16323 compcco 16324 ×_c cxpc 17168

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336

This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-hom 16336 df-cco 16337 df-xpc 17172

This theorem is referenced by: xpchomfval 17179 xpccofval 17182 xpchom2 17186 xpcco2 17187 xpccatid 17188 1stfval 17191 2ndfval 17194 1stfcl 17197 2ndfcl 17198 prfcl 17203 prf1st 17204 prf2nd 17205 1st2ndprf 17206 catcxpccl 17207 xpcpropd 17208 evlfcl 17222 curf1cl 17228 curf2cl 17231 curfcl 17232 uncf1 17236 uncf2 17237 uncfcurf 17239 diag11 17243 diag12 17244 diag2 17245 curf2ndf 17247 hofcl 17259 yonedalem21 17273 yonedalem22 17278 yonedalem3b 17279 yonedalem3 17280 yonedainv 17281 yonffthlem 17282

W3C validator