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Theorem xpcbas 16818

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 2622	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
5		eqid 2622	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
6		eqid 2622	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
7		eqid 2622	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
8		simpl 473	. . . . 5 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐶 ∈ V)
9		simpr 477	. . . . 5 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐷 ∈ V)
10		eqidd 2623	. . . . 5 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = (𝑋 × 𝑌))
11		eqidd 2623	. . . . 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 2623	. . . . 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 16817	. . . 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 16617	. . . 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 15919	. . . 4 ⊢ Base = Slot (Base‘ndx)
16		snsstp1 4347	. . . 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		fvex 6201	. . . . . . 7 ⊢ (Base‘𝐶) ∈ V
18	2, 17	eqeltri 2697	. . . . . 6 ⊢ 𝑋 ∈ V
19		fvex 6201	. . . . . . 7 ⊢ (Base‘𝐷) ∈ V
20	3, 19	eqeltri 2697	. . . . . 6 ⊢ 𝑌 ∈ V
21	18, 20	xpex 6962	. . . . 5 ⊢ (𝑋 × 𝑌) ∈ V
22	21	a1i 11	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) ∈ V)
23		eqid 2622	. . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇)
24	13, 14, 15, 16, 22, 23	strfv3 15908	. . 3 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (Base‘𝑇) = (𝑋 × 𝑌))
25	24	eqcomd 2628	. 2 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = (Base‘𝑇))
26		base0 15912	. . 3 ⊢ ∅ = (Base‘∅)
27		fvprc 6185	. . . . . 6 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = ∅)
28	2, 27	syl5eq 2668	. . . . 5 ⊢ (¬ 𝐶 ∈ V → 𝑋 = ∅)
29		fvprc 6185	. . . . . 6 ⊢ (¬ 𝐷 ∈ V → (Base‘𝐷) = ∅)
30	3, 29	syl5eq 2668	. . . . 5 ⊢ (¬ 𝐷 ∈ V → 𝑌 = ∅)
31	28, 30	orim12i 538	. . . 4 ⊢ ((¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V) → (𝑋 = ∅ ∨ 𝑌 = ∅))
32		ianor 509	. . . 4 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V))
33		xpeq0 5554	. . . 4 ⊢ ((𝑋 × 𝑌) = ∅ ↔ (𝑋 = ∅ ∨ 𝑌 = ∅))
34	31, 32, 33	3imtr4i 281	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = ∅)
35		fnxpc 16816	. . . . . . 7 ⊢ ×_c Fn (V × V)
36		fndm 5990	. . . . . . 7 ⊢ ( ×_c Fn (V × V) → dom ×_c = (V × V))
37	35, 36	ax-mp 5	. . . . . 6 ⊢ dom ×_c = (V × V)
38	37	ndmov 6818	. . . . 5 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 ×_c 𝐷) = ∅)
39	1, 38	syl5eq 2668	. . . 4 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑇 = ∅)
40	39	fveq2d 6195	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (Base‘𝑇) = (Base‘∅))
41	26, 34, 40	3eqtr4a 2682	. 2 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = (Base‘𝑇))
42	25, 41	pm2.61i 176	1 ⊢ (𝑋 × 𝑌) = (Base‘𝑇)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 {ctp 4181 ⟨cop 4183 × cxp 5112 dom cdm 5114 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 1^st c1st 7166 2^nd c2nd 7167 1c1 9937 5c5 11073 cdc 11493 ndxcnx 15854 Basecbs 15857 Hom chom 15952 compcco 15953 ×_c cxpc 16808

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-hom 15966 df-cco 15967 df-xpc 16812

This theorem is referenced by: xpchomfval 16819 xpccofval 16822 xpchom2 16826 xpcco2 16827 xpccatid 16828 1stfval 16831 2ndfval 16834 1stfcl 16837 2ndfcl 16838 prfcl 16843 prf1st 16844 prf2nd 16845 1st2ndprf 16846 catcxpccl 16847 xpcpropd 16848 evlfcl 16862 curf1cl 16868 curf2cl 16871 curfcl 16872 uncf1 16876 uncf2 16877 uncfcurf 16879 diag11 16883 diag12 16884 diag2 16885 curf2ndf 16887 hofcl 16899 yonedalem21 16913 yonedalem22 16918 yonedalem3b 16919 yonedalem3 16920 yonedainv 16921 yonffthlem 16922

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