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Theorem xpsff1o 16614

Description: The function appearing in xpsval 16618 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 16611	. . . . . 6 ⊢ (^◡({𝑥} +_𝑐 {𝑦}) ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
2	1	biimpri 220	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ^◡({𝑥} +_𝑐 {𝑦}) ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵))
3	2	rgen2 3157	. . . 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 7517	. . . 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 222	. . 3 ⊢ 𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)
7		1st2nd2 7484	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
8	7	fveq2d 6450	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘𝑧) = (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩))
9		df-ov 6925	. . . . . . . 8 ⊢ ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)) = (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
10		xp1st 7477	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (1^st ‘𝑧) ∈ 𝐴)
11		xp2nd 7478	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (2^nd ‘𝑧) ∈ 𝐵)
12	4	xpsfval 16613	. . . . . . . . 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 579	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)) = ^◡({(1^st ‘𝑧)} +_𝑐 {(2^nd ‘𝑧)}))
14	9, 13	syl5eqr 2828	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩) = ^◡({(1^st ‘𝑧)} +_𝑐 {(2^nd ‘𝑧)}))
15	8, 14	eqtrd 2814	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘𝑧) = ^◡({(1^st ‘𝑧)} +_𝑐 {(2^nd ‘𝑧)}))
16		1st2nd2 7484	. . . . . . . 8 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → 𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
17	16	fveq2d 6450	. . . . . . 7 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘𝑤) = (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩))
18		df-ov 6925	. . . . . . . 8 ⊢ ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)) = (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
19		xp1st 7477	. . . . . . . . 9 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (1^st ‘𝑤) ∈ 𝐴)
20		xp2nd 7478	. . . . . . . . 9 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (2^nd ‘𝑤) ∈ 𝐵)
21	4	xpsfval 16613	. . . . . . . . 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 579	. . . . . . . 8 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)) = ^◡({(1^st ‘𝑤)} +_𝑐 {(2^nd ‘𝑤)}))
23	18, 22	syl5eqr 2828	. . . . . . 7 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩) = ^◡({(1^st ‘𝑤)} +_𝑐 {(2^nd ‘𝑤)}))
24	17, 23	eqtrd 2814	. . . . . 6 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘𝑤) = ^◡({(1^st ‘𝑤)} +_𝑐 {(2^nd ‘𝑤)}))
25	15, 24	eqeqan12d 2794	. . . . 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 6445	. . . . . . . 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 6459	. . . . . . . . 9 ⊢ (1^st ‘𝑧) ∈ V
28		xpsc0 16606	. . . . . . . . 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 6459	. . . . . . . . 9 ⊢ (1^st ‘𝑤) ∈ V
31		xpsc0 16606	. . . . . . . . 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 2837	. . . . . . 7 ⊢ (^◡({(1^st ‘𝑧)} +_𝑐 {(2^nd ‘𝑧)}) = ^◡({(1^st ‘𝑤)} +_𝑐 {(2^nd ‘𝑤)}) → (1^st ‘𝑧) = (1^st ‘𝑤))
34		fveq1 6445	. . . . . . . 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 6459	. . . . . . . . 9 ⊢ (2^nd ‘𝑧) ∈ V
36		xpsc1 16607	. . . . . . . . 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 6459	. . . . . . . . 9 ⊢ (2^nd ‘𝑤) ∈ V
39		xpsc1 16607	. . . . . . . . 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 2837	. . . . . . 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 4644	. . . . . 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 2794	. . . . . 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 238	. . . . 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 232	. . . 4 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
46	45	rgen2 3157	. . 3 ⊢ ∀𝑧 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)
47		dff13 6784	. . 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 701	. 2 ⊢ 𝐹:(𝐴 × 𝐵)–1-1→X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)
49		xpsfrnel 16609	. . . . . 6 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑧 Fn 2_o ∧ (𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1_o) ∈ 𝐵))
50	49	simp2bi 1137	. . . . 5 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘∅) ∈ 𝐴)
51	49	simp3bi 1138	. . . . 5 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘1_o) ∈ 𝐵)
52	4	xpsfval 16613	. . . . . . 7 ⊢ (((𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1_o) ∈ 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1_o)) = ^◡({(𝑧‘∅)} +_𝑐 {(𝑧‘1_o)}))
53	50, 51, 52	syl2anc 579	. . . . . 6 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1_o)) = ^◡({(𝑧‘∅)} +_𝑐 {(𝑧‘1_o)}))
54		ixpfn 8200	. . . . . . 7 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 Fn 2_o)
55		xpsfeq 16610	. . . . . . 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 2815	. . . . 5 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 = ((𝑧‘∅)𝐹(𝑧‘1_o)))
58		rspceov 6968	. . . . 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 1439	. . . 4 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎𝐹𝑏))
60	59	rgen 3104	. . 3 ⊢ ∀𝑧 ∈ X 𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎𝐹𝑏)
61		foov 7085	. . 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 701	. 2 ⊢ 𝐹:(𝐴 × 𝐵)–onto→X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)
63		df-f1o 6142	. 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 701	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 386 = wceq 1601 ∈ wcel 2107 ∀wral 3090 ∃wrex 3091 Vcvv 3398 ∅c0 4141 ifcif 4307 {csn 4398 ⟨cop 4404 × cxp 5353 ^◡ccnv 5354 Fn wfn 6130 ⟶wf 6131 –1-1→wf1 6132 –onto→wfo 6133 –1-1-onto→wf1o 6134 ‘cfv 6135 (class class class)co 6922 ↦ cmpt2 6924 1^st c1st 7443 2^nd c2nd 7444 1_oc1o 7836 2_oc2o 7837 Xcixp 8194 +_𝑐 ccda 9324

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-cda 9325

This theorem is referenced by: xpsfrn 16615 xpsff1o2 16617

W3C validator