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Theorem xpsff1o 17195

Description: The function appearing in xpsval 17198 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⟩, ⟨1_o, 𝑦⟩})

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

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

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

Step	Hyp	Ref	Expression
1		xpsfrnel2 17192	. . . . . 6 ⊢ ({⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩} ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
2	1	biimpri 227	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩} ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵))
3	2	rgen2 3126	. . . 4 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩} ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)
4		xpsff1o.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩})
5	4	fmpo 7881	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩} ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) ↔ 𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵))
6	3, 5	mpbi 229	. . 3 ⊢ 𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)
7		1st2nd2 7843	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
8	7	fveq2d 6760	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘𝑧) = (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩))
9		df-ov 7258	. . . . . . . 8 ⊢ ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)) = (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
10		xp1st 7836	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (1^st ‘𝑧) ∈ 𝐴)
11		xp2nd 7837	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (2^nd ‘𝑧) ∈ 𝐵)
12	4	xpsfval 17194	. . . . . . . . 9 ⊢ (((1^st ‘𝑧) ∈ 𝐴 ∧ (2^nd ‘𝑧) ∈ 𝐵) → ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)) = {⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩})
13	10, 11, 12	syl2anc 583	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)) = {⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩})
14	9, 13	eqtr3id 2793	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩) = {⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩})
15	8, 14	eqtrd 2778	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘𝑧) = {⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩})
16		1st2nd2 7843	. . . . . . . 8 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → 𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
17	16	fveq2d 6760	. . . . . . 7 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘𝑤) = (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩))
18		df-ov 7258	. . . . . . . 8 ⊢ ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)) = (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
19		xp1st 7836	. . . . . . . . 9 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (1^st ‘𝑤) ∈ 𝐴)
20		xp2nd 7837	. . . . . . . . 9 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (2^nd ‘𝑤) ∈ 𝐵)
21	4	xpsfval 17194	. . . . . . . . 9 ⊢ (((1^st ‘𝑤) ∈ 𝐴 ∧ (2^nd ‘𝑤) ∈ 𝐵) → ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)) = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩})
22	19, 20, 21	syl2anc 583	. . . . . . . 8 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)) = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩})
23	18, 22	eqtr3id 2793	. . . . . . 7 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩) = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩})
24	17, 23	eqtrd 2778	. . . . . 6 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘𝑤) = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩})
25	15, 24	eqeqan12d 2752	. . . . 5 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ {⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩} = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩}))
26		fveq1 6755	. . . . . . . 8 ⊢ ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩} = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩} → ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩}‘∅) = ({⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩}‘∅))
27		fvex 6769	. . . . . . . . 9 ⊢ (1^st ‘𝑧) ∈ V
28		fvpr0o 17187	. . . . . . . . 9 ⊢ ((1^st ‘𝑧) ∈ V → ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩}‘∅) = (1^st ‘𝑧))
29	27, 28	ax-mp 5	. . . . . . . 8 ⊢ ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩}‘∅) = (1^st ‘𝑧)
30		fvex 6769	. . . . . . . . 9 ⊢ (1^st ‘𝑤) ∈ V
31		fvpr0o 17187	. . . . . . . . 9 ⊢ ((1^st ‘𝑤) ∈ V → ({⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩}‘∅) = (1^st ‘𝑤))
32	30, 31	ax-mp 5	. . . . . . . 8 ⊢ ({⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩}‘∅) = (1^st ‘𝑤)
33	26, 29, 32	3eqtr3g 2802	. . . . . . 7 ⊢ ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩} = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩} → (1^st ‘𝑧) = (1^st ‘𝑤))
34		fveq1 6755	. . . . . . . 8 ⊢ ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩} = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩} → ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩}‘1_o) = ({⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩}‘1_o))
35		fvex 6769	. . . . . . . . 9 ⊢ (2^nd ‘𝑧) ∈ V
36		fvpr1o 17188	. . . . . . . . 9 ⊢ ((2^nd ‘𝑧) ∈ V → ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩}‘1_o) = (2^nd ‘𝑧))
37	35, 36	ax-mp 5	. . . . . . . 8 ⊢ ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩}‘1_o) = (2^nd ‘𝑧)
38		fvex 6769	. . . . . . . . 9 ⊢ (2^nd ‘𝑤) ∈ V
39		fvpr1o 17188	. . . . . . . . 9 ⊢ ((2^nd ‘𝑤) ∈ V → ({⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩}‘1_o) = (2^nd ‘𝑤))
40	38, 39	ax-mp 5	. . . . . . . 8 ⊢ ({⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩}‘1_o) = (2^nd ‘𝑤)
41	34, 37, 40	3eqtr3g 2802	. . . . . . 7 ⊢ ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩} = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩} → (2^nd ‘𝑧) = (2^nd ‘𝑤))
42	33, 41	opeq12d 4809	. . . . . 6 ⊢ ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩} = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩} → ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
43	7, 16	eqeqan12d 2752	. . . . . 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 245	. . . . 5 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩} = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩} → 𝑧 = 𝑤))
45	25, 44	sylbid 239	. . . 4 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
46	45	rgen2 3126	. . 3 ⊢ ∀𝑧 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)
47		dff13 7109	. . 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 707	. 2 ⊢ 𝐹:(𝐴 × 𝐵)–1-1→X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)
49		xpsfrnel 17190	. . . . . 6 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑧 Fn 2_o ∧ (𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1_o) ∈ 𝐵))
50	49	simp2bi 1144	. . . . 5 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘∅) ∈ 𝐴)
51	49	simp3bi 1145	. . . . 5 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘1_o) ∈ 𝐵)
52	4	xpsfval 17194	. . . . . . 7 ⊢ (((𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1_o) ∈ 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1_o)) = {⟨∅, (𝑧‘∅)⟩, ⟨1_o, (𝑧‘1_o)⟩})
53	50, 51, 52	syl2anc 583	. . . . . 6 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1_o)) = {⟨∅, (𝑧‘∅)⟩, ⟨1_o, (𝑧‘1_o)⟩})
54		ixpfn 8649	. . . . . . 7 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 Fn 2_o)
55		xpsfeq 17191	. . . . . . 7 ⊢ (𝑧 Fn 2_o → {⟨∅, (𝑧‘∅)⟩, ⟨1_o, (𝑧‘1_o)⟩} = 𝑧)
56	54, 55	syl 17	. . . . . 6 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → {⟨∅, (𝑧‘∅)⟩, ⟨1_o, (𝑧‘1_o)⟩} = 𝑧)
57	53, 56	eqtr2d 2779	. . . . 5 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 = ((𝑧‘∅)𝐹(𝑧‘1_o)))
58		rspceov 7302	. . . . 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 1369	. . . 4 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎𝐹𝑏))
60	59	rgen 3073	. . 3 ⊢ ∀𝑧 ∈ X 𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎𝐹𝑏)
61		foov 7424	. . 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 707	. 2 ⊢ 𝐹:(𝐴 × 𝐵)–onto→X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)
63		df-f1o 6425	. 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 707	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 395 = wceq 1539 ∈ wcel 2108 ∀wral 3063 ∃wrex 3064 Vcvv 3422 ∅c0 4253 ifcif 4456 {cpr 4560 ⟨cop 4564 × cxp 5578 Fn wfn 6413 ⟶wf 6414 –1-1→wf1 6415 –onto→wfo 6416 –1-1-onto→wf1o 6417 ‘cfv 6418 (class class class)co 7255 ∈ cmpo 7257 1^st c1st 7802 2^nd c2nd 7803 1_oc1o 8260 2_oc2o 8261 Xcixp 8643

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-1o 8267 df-2o 8268 df-ixp 8644 df-en 8692 df-fin 8695

This theorem is referenced by: xpsfrn 17196 xpsff1o2 17197

W3C validator