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Theorem xpsff1o 16051

Description: The function appearing in xpsval 16055 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair 2_𝑜 = {∅, 1_𝑜}. (Contributed by Mario Carneiro, 15-Aug-2015.)

**Hypothesis**
Ref	Expression
xpsff1o.f	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ^◡({𝑥} +_𝑐 {𝑦}))

**Assertion**
Ref	Expression
xpsff1o	⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→X𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵)

Distinct variable groups: 𝑥,𝑘,𝑦,𝐴 𝐵,𝑘,𝑥,𝑦 Allowed substitution hints: 𝐹(𝑥,𝑦,𝑘)

Proof of Theorem xpsff1o Dummy variables 𝑎 𝑏 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpsfrnel2 16048	. . . . . 6 ⊢ (^◡({𝑥} +_𝑐 {𝑦}) ∈ X𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
2	1	biimpri 217	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ^◡({𝑥} +_𝑐 {𝑦}) ∈ X𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵))
3	2	rgen2 2958	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ^◡({𝑥} +_𝑐 {𝑦}) ∈ X𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵)
4		xpsff1o.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ^◡({𝑥} +_𝑐 {𝑦}))
5	4	fmpt2 7126	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ^◡({𝑥} +_𝑐 {𝑦}) ∈ X𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ 𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵))
6	3, 5	mpbi 219	. . 3 ⊢ 𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵)
7		1st2nd2 7096	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
8	7	fveq2d 6107	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘𝑧) = (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩))
9		df-ov 6552	. . . . . . . 8 ⊢ ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)) = (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
10		xp1st 7089	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (1^st ‘𝑧) ∈ 𝐴)
11		xp2nd 7090	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (2^nd ‘𝑧) ∈ 𝐵)
12	4	xpsfval 16050	. . . . . . . . 9 ⊢ (((1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) → ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)) = ^◡({(1^st ‘𝑧)} +_𝑐 {(2^nd ‘𝑧)}))
13	10, 11, 12	syl2anc 691	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)) = ^◡({(1^st ‘𝑧)} +_𝑐 {(2^nd ‘𝑧)}))
14	9, 13	syl5eqr 2658	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩) = ^◡({(1^st ‘𝑧)} +_𝑐 {(2^nd ‘𝑧)}))
15	8, 14	eqtrd 2644	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘𝑧) = ^◡({(1^st ‘𝑧)} +_𝑐 {(2^nd ‘𝑧)}))
16		1st2nd2 7096	. . . . . . . 8 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → 𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
17	16	fveq2d 6107	. . . . . . 7 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘𝑤) = (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩))
18		df-ov 6552	. . . . . . . 8 ⊢ ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)) = (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
19		xp1st 7089	. . . . . . . . 9 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (1^st ‘𝑤) ∈ 𝐴)
20		xp2nd 7090	. . . . . . . . 9 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (2^nd ‘𝑤) ∈ 𝐵)
21	4	xpsfval 16050	. . . . . . . . 9 ⊢ (((1^st ‘𝑤) ∈ 𝐴 ∧ (2^nd ‘𝑤) ∈ 𝐵) → ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)) = ^◡({(1^st ‘𝑤)} +_𝑐 {(2^nd ‘𝑤)}))
22	19, 20, 21	syl2anc 691	. . . . . . . 8 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)) = ^◡({(1^st ‘𝑤)} +_𝑐 {(2^nd ‘𝑤)}))
23	18, 22	syl5eqr 2658	. . . . . . 7 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩) = ^◡({(1^st ‘𝑤)} +_𝑐 {(2^nd ‘𝑤)}))
24	17, 23	eqtrd 2644	. . . . . 6 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘𝑤) = ^◡({(1^st ‘𝑤)} +_𝑐 {(2^nd ‘𝑤)}))
25	15, 24	eqeqan12d 2626	. . . . 5 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ^◡({(1^st ‘𝑧)} +_𝑐 {(2^nd ‘𝑧)}) = ^◡({(1^st ‘𝑤)} +_𝑐 {(2^nd ‘𝑤)})))
26		fveq1 6102	. . . . . . . 8 ⊢ (^◡({(1^st ‘𝑧)} +_𝑐 {(2^nd ‘𝑧)}) = ^◡({(1^st ‘𝑤)} +_𝑐 {(2^nd ‘𝑤)}) → (^◡({(1^st ‘𝑧)} +_𝑐 {(2^nd ‘𝑧)})‘∅) = (^◡({(1^st ‘𝑤)} +_𝑐 {(2^nd ‘𝑤)})‘∅))
27		fvex 6113	. . . . . . . . 9 ⊢ (1^st ‘𝑧) ∈ V
28		xpsc0 16043	. . . . . . . . 9 ⊢ ((1^st ‘𝑧) ∈ V → (^◡({(1^st ‘𝑧)} +_𝑐 {(2^nd ‘𝑧)})‘∅) = (1^st ‘𝑧))
29	27, 28	ax-mp 5	. . . . . . . 8 ⊢ (^◡({(1^st ‘𝑧)} +_𝑐 {(2^nd ‘𝑧)})‘∅) = (1^st ‘𝑧)
30		fvex 6113	. . . . . . . . 9 ⊢ (1^st ‘𝑤) ∈ V
31		xpsc0 16043	. . . . . . . . 9 ⊢ ((1^st ‘𝑤) ∈ V → (^◡({(1^st ‘𝑤)} +_𝑐 {(2^nd ‘𝑤)})‘∅) = (1^st ‘𝑤))
32	30, 31	ax-mp 5	. . . . . . . 8 ⊢ (^◡({(1^st ‘𝑤)} +_𝑐 {(2^nd ‘𝑤)})‘∅) = (1^st ‘𝑤)
33	26, 29, 32	3eqtr3g 2667	. . . . . . 7 ⊢ (^◡({(1^st ‘𝑧)} +_𝑐 {(2^nd ‘𝑧)}) = ^◡({(1^st ‘𝑤)} +_𝑐 {(2^nd ‘𝑤)}) → (1^st ‘𝑧) = (1^st ‘𝑤))
34		fveq1 6102	. . . . . . . 8 ⊢ (^◡({(1^st ‘𝑧)} +_𝑐 {(2^nd ‘𝑧)}) = ^◡({(1^st ‘𝑤)} +_𝑐 {(2^nd ‘𝑤)}) → (^◡({(1^st ‘𝑧)} +_𝑐 {(2^nd ‘𝑧)})‘1_𝑜) = (^◡({(1^st ‘𝑤)} +_𝑐 {(2^nd ‘𝑤)})‘1_𝑜))
35		fvex 6113	. . . . . . . . 9 ⊢ (2^nd ‘𝑧) ∈ V
36		xpsc1 16044	. . . . . . . . 9 ⊢ ((2^nd ‘𝑧) ∈ V → (^◡({(1^st ‘𝑧)} +_𝑐 {(2^nd ‘𝑧)})‘1_𝑜) = (2^nd ‘𝑧))
37	35, 36	ax-mp 5	. . . . . . . 8 ⊢ (^◡({(1^st ‘𝑧)} +_𝑐 {(2^nd ‘𝑧)})‘1_𝑜) = (2^nd ‘𝑧)
38		fvex 6113	. . . . . . . . 9 ⊢ (2^nd ‘𝑤) ∈ V
39		xpsc1 16044	. . . . . . . . 9 ⊢ ((2^nd ‘𝑤) ∈ V → (^◡({(1^st ‘𝑤)} +_𝑐 {(2^nd ‘𝑤)})‘1_𝑜) = (2^nd ‘𝑤))
40	38, 39	ax-mp 5	. . . . . . . 8 ⊢ (^◡({(1^st ‘𝑤)} +_𝑐 {(2^nd ‘𝑤)})‘1_𝑜) = (2^nd ‘𝑤)
41	34, 37, 40	3eqtr3g 2667	. . . . . . 7 ⊢ (^◡({(1^st ‘𝑧)} +_𝑐 {(2^nd ‘𝑧)}) = ^◡({(1^st ‘𝑤)} +_𝑐 {(2^nd ‘𝑤)}) → (2^nd ‘𝑧) = (2^nd ‘𝑤))
42	33, 41	opeq12d 4348	. . . . . 6 ⊢ (^◡({(1^st ‘𝑧)} +_𝑐 {(2^nd ‘𝑧)}) = ^◡({(1^st ‘𝑤)} +_𝑐 {(2^nd ‘𝑤)}) → ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
43	7, 16	eqeqan12d 2626	. . . . . 6 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → (𝑧 = 𝑤 ↔ ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩))
44	42, 43	syl5ibr 235	. . . . 5 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → (^◡({(1^st ‘𝑧)} +_𝑐 {(2^nd ‘𝑧)}) = ^◡({(1^st ‘𝑤)} +_𝑐 {(2^nd ‘𝑤)}) → 𝑧 = 𝑤))
45	25, 44	sylbid 229	. . . 4 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
46	45	rgen2 2958	. . 3 ⊢ ∀𝑧 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)
47		dff13 6416	. . 3 ⊢ (𝐹:(𝐴 × 𝐵)–1-1→X𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ∧ ∀𝑧 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)))
48	6, 46, 47	mpbir2an 957	. 2 ⊢ 𝐹:(𝐴 × 𝐵)–1-1→X𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵)
49		xpsfrnel 16046	. . . . . 6 ⊢ (𝑧 ∈ X𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑧 Fn 2_𝑜 ∧ (𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1_𝑜) ∈ 𝐵))
50	49	simp2bi 1070	. . . . 5 ⊢ (𝑧 ∈ X𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘∅) ∈ 𝐴)
51	49	simp3bi 1071	. . . . 5 ⊢ (𝑧 ∈ X𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘1_𝑜) ∈ 𝐵)
52	4	xpsfval 16050	. . . . . . 7 ⊢ (((𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1_𝑜) ∈ 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1_𝑜)) = ^◡({(𝑧‘∅)} +_𝑐 {(𝑧‘1_𝑜)}))
53	50, 51, 52	syl2anc 691	. . . . . 6 ⊢ (𝑧 ∈ X𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1_𝑜)) = ^◡({(𝑧‘∅)} +_𝑐 {(𝑧‘1_𝑜)}))
54		ixpfn 7800	. . . . . . 7 ⊢ (𝑧 ∈ X𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 Fn 2_𝑜)
55		xpsfeq 16047	. . . . . . 7 ⊢ (𝑧 Fn 2_𝑜 → ^◡({(𝑧‘∅)} +_𝑐 {(𝑧‘1_𝑜)}) = 𝑧)
56	54, 55	syl 17	. . . . . 6 ⊢ (𝑧 ∈ X𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → ^◡({(𝑧‘∅)} +_𝑐 {(𝑧‘1_𝑜)}) = 𝑧)
57	53, 56	eqtr2d 2645	. . . . 5 ⊢ (𝑧 ∈ X𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 = ((𝑧‘∅)𝐹(𝑧‘1_𝑜)))
58		rspceov 6590	. . . . 5 ⊢ (((𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1_𝑜) ∈ 𝐵 ∧ 𝑧 = ((𝑧‘∅)𝐹(𝑧‘1_𝑜))) → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎𝐹𝑏))
59	50, 51, 57, 58	syl3anc 1318	. . . 4 ⊢ (𝑧 ∈ X𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎𝐹𝑏))
60	59	rgen 2906	. . 3 ⊢ ∀𝑧 ∈ X 𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵)∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎𝐹𝑏)
61		foov 6706	. . 3 ⊢ (𝐹:(𝐴 × 𝐵)–onto→X𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ∧ ∀𝑧 ∈ X 𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵)∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎𝐹𝑏)))
62	6, 60, 61	mpbir2an 957	. 2 ⊢ 𝐹:(𝐴 × 𝐵)–onto→X𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵)
63		df-f1o 5811	. 2 ⊢ (𝐹:(𝐴 × 𝐵)–1-1-onto→X𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐹:(𝐴 × 𝐵)–1-1→X𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ∧ 𝐹:(𝐴 × 𝐵)–onto→X𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵)))
64	48, 62, 63	mpbir2an 957	1 ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→X𝑘 ∈ 2_𝑜 if(𝑘 = ∅, 𝐴, 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ∅c0 3874 ifcif 4036 {csn 4125 ⟨cop 4131 × cxp 5036 ^◡ccnv 5037 Fn wfn 5799 ⟶wf 5800 –1-1→wf1 5801 –onto→wfo 5802 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1^st c1st 7057 2^nd c2nd 7058 1_𝑜c1o 7440 2_𝑜c2o 7441 Xcixp 7794 +_𝑐 ccda 8872

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-cda 8873

This theorem is referenced by: xpsfrn 16052 xpsff1o2 16054

W3C validator