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Theorem xpcbas 18224

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	. . . 4 ⊢ 𝑇 = (𝐶 ×_c 𝐷)
2		xpcbas.x	. . . 4 ⊢ 𝑋 = (Base‘𝐶)
3		xpcbas.y	. . . 4 ⊢ 𝑌 = (Base‘𝐷)
4		eqid 2765	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
5		eqid 2765	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
6		eqid 2765	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
7		eqid 2765	. . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷)
8		simpl 487	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐶 ∈ V)
9		simpr 489	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐷 ∈ V)
10		eqidd 2766	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = (𝑋 × 𝑌))
11		eqidd 2766	. . . 4 ⊢ ((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 2766	. . . 4 ⊢ ((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 18223	. . 3 ⊢ ((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	2	fvexi 6885	. . . . 5 ⊢ 𝑋 ∈ V
15	3	fvexi 6885	. . . . 5 ⊢ 𝑌 ∈ V
16	14, 15	xpex 7740	. . . 4 ⊢ (𝑋 × 𝑌) ∈ V
17	16	a1i 11	. . 3 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) ∈ V)
18	13, 17	estrreslem1 18183	. 2 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = (Base‘𝑇))
19		base0 17264	. . 3 ⊢ ∅ = (Base‘∅)
20		fvprc 6863	. . . . . 6 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = ∅)
21	2, 20	eqtrid 2812	. . . . 5 ⊢ (¬ 𝐶 ∈ V → 𝑋 = ∅)
22		fvprc 6863	. . . . . 6 ⊢ (¬ 𝐷 ∈ V → (Base‘𝐷) = ∅)
23	3, 22	eqtrid 2812	. . . . 5 ⊢ (¬ 𝐷 ∈ V → 𝑌 = ∅)
24	21, 23	orim12i 921	. . . 4 ⊢ ((¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V) → (𝑋 = ∅ ∨ 𝑌 = ∅))
25		ianor 997	. . . 4 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V))
26		xpeq0 6149	. . . 4 ⊢ ((𝑋 × 𝑌) = ∅ ↔ (𝑋 = ∅ ∨ 𝑌 = ∅))
27	24, 25, 26	3imtr4i 295	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = ∅)
28		fnxpc 18222	. . . . . . 7 ⊢ ×_c Fn (V × V)
29		fndm 6628	. . . . . . 7 ⊢ ( ×_c Fn (V × V) → dom ×_c = (V × V))
30	28, 29	ax-mp 5	. . . . . 6 ⊢ dom ×_c = (V × V)
31	30	ndmov 7584	. . . . 5 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 ×_c 𝐷) = ∅)
32	1, 31	eqtrid 2812	. . . 4 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑇 = ∅)
33	32	fveq2d 6875	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (Base‘𝑇) = (Base‘∅))
34	19, 27, 33	3eqtr4a 2826	. 2 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = (Base‘𝑇))
35	18, 34	pm2.61i 184	1 ⊢ (𝑋 × 𝑌) = (Base‘𝑇)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 400 ∨ wo 860 = wceq 1563 ∈ wcel 2145 Vcvv 3457 ∅c0 4288 ⟨cop 4591 × cxp 5650 dom cdm 5652 Fn wfn 6520 ‘cfv 6525 (class class class)co 7400 ∈ cmpo 7402 1^st c1st 7972 2^nd c2nd 7973 Basecbs 17259 Hom chom 17311 compcco 17312 ×_c cxpc 18214

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-om 7851 df-1st 7974 df-2nd 7975 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-er 8682 df-en 8932 df-dom 8933 df-sdom 8934 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-z 12583 df-dec 12703 df-slot 17232 df-ndx 17244 df-base 17260 df-hom 17324 df-cco 17325 df-xpc 18218

This theorem is referenced by: xpchomfval 18225 xpccofval 18228 xpchom2 18232 xpcco2 18233 xpccatid 18234 1stfval 18237 2ndfval 18240 1stfcl 18243 2ndfcl 18244 prfcl 18249 prf1st 18250 prf2nd 18251 1st2ndprf 18252 catcxpccl 18253 xpcpropd 18254 evlfcl 18268 curf1cl 18274 curf2cl 18277 curfcl 18278 uncf1 18282 uncf2 18283 uncfcurf 18285 diag11 18289 diag12 18290 diag2 18291 curf2ndf 18293 hofcl 18305 yonedalem21 18319 yonedalem22 18324 yonedalem3b 18325 yonedalem3 18326 yonedainv 18327 yonffthlem 18328 elxpcbasex1ALT 49878 elxpcbasex2ALT 49880 xpcfucbas 49881 dfswapf2 49890 swapf1a 49898 swapf1 49901 swapf2val 49902 swapf1f1o 49904 swapf2f1oa 49906 swapfida 49909 oppc1stf 49917 oppc2ndf 49918 cofuswapf1 49923 cofuswapf2 49924

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