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Theorem xpsff1o 17613

Description: The function appearing in xpsval 17616 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 17610	. . . . . 6 ⊢ ({⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩} ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
2	1	biimpri 228	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩} ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵))
3	2	rgen2 3198	. . . 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 8094	. . . 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 230	. . 3 ⊢ 𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)
7		1st2nd2 8054	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
8	7	fveq2d 6909	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘𝑧) = (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩))
9		df-ov 7435	. . . . . . . 8 ⊢ ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)) = (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
10		xp1st 8047	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (1^st ‘𝑧) ∈ 𝐴)
11		xp2nd 8048	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (2^nd ‘𝑧) ∈ 𝐵)
12	4	xpsfval 17612	. . . . . . . . 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 584	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)) = {⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩})
14	9, 13	eqtr3id 2790	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩) = {⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩})
15	8, 14	eqtrd 2776	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘𝑧) = {⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩})
16		1st2nd2 8054	. . . . . . . 8 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → 𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
17	16	fveq2d 6909	. . . . . . 7 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘𝑤) = (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩))
18		df-ov 7435	. . . . . . . 8 ⊢ ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)) = (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
19		xp1st 8047	. . . . . . . . 9 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (1^st ‘𝑤) ∈ 𝐴)
20		xp2nd 8048	. . . . . . . . 9 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (2^nd ‘𝑤) ∈ 𝐵)
21	4	xpsfval 17612	. . . . . . . . 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 584	. . . . . . . 8 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)) = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩})
23	18, 22	eqtr3id 2790	. . . . . . 7 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩) = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩})
24	17, 23	eqtrd 2776	. . . . . 6 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘𝑤) = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩})
25	15, 24	eqeqan12d 2750	. . . . 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 6904	. . . . . . . 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 6918	. . . . . . . . 9 ⊢ (1^st ‘𝑧) ∈ V
28		fvpr0o 17605	. . . . . . . . 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 6918	. . . . . . . . 9 ⊢ (1^st ‘𝑤) ∈ V
31		fvpr0o 17605	. . . . . . . . 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 2799	. . . . . . 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 6904	. . . . . . . 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 6918	. . . . . . . . 9 ⊢ (2^nd ‘𝑧) ∈ V
36		fvpr1o 17606	. . . . . . . . 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 6918	. . . . . . . . 9 ⊢ (2^nd ‘𝑤) ∈ V
39		fvpr1o 17606	. . . . . . . . 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 2799	. . . . . . 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 4880	. . . . . 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 2750	. . . . . 6 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → (𝑧 = 𝑤 ↔ ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩))
44	42, 43	imbitrrid 246	. . . . 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 240	. . . 4 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
46	45	rgen2 3198	. . 3 ⊢ ∀𝑧 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)
47		dff13 7276	. . 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 711	. 2 ⊢ 𝐹:(𝐴 × 𝐵)–1-1→X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)
49		xpsfrnel 17608	. . . . . 6 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑧 Fn 2_o ∧ (𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1_o) ∈ 𝐵))
50	49	simp2bi 1146	. . . . 5 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘∅) ∈ 𝐴)
51	49	simp3bi 1147	. . . . 5 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘1_o) ∈ 𝐵)
52	4	xpsfval 17612	. . . . . . 7 ⊢ (((𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1_o) ∈ 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1_o)) = {⟨∅, (𝑧‘∅)⟩, ⟨1_o, (𝑧‘1_o)⟩})
53	50, 51, 52	syl2anc 584	. . . . . 6 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1_o)) = {⟨∅, (𝑧‘∅)⟩, ⟨1_o, (𝑧‘1_o)⟩})
54		ixpfn 8944	. . . . . . 7 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 Fn 2_o)
55		xpsfeq 17609	. . . . . . 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 2777	. . . . 5 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 = ((𝑧‘∅)𝐹(𝑧‘1_o)))
58		rspceov 7481	. . . . 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 1372	. . . 4 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎𝐹𝑏))
60	59	rgen 3062	. . 3 ⊢ ∀𝑧 ∈ X 𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎𝐹𝑏)
61		foov 7608	. . 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 711	. 2 ⊢ 𝐹:(𝐴 × 𝐵)–onto→X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)
63		df-f1o 6567	. 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 711	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 2107 ∀wral 3060 ∃wrex 3069 Vcvv 3479 ∅c0 4332 ifcif 4524 {cpr 4627 ⟨cop 4631 × cxp 5682 Fn wfn 6555 ⟶wf 6556 –1-1→wf1 6557 –onto→wfo 6558 –1-1-onto→wf1o 6559 ‘cfv 6560 (class class class)co 7432 ∈ cmpo 7434 1^st c1st 8013 2^nd c2nd 8014 1_oc1o 8500 2_oc2o 8501 Xcixp 8938

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-1o 8507 df-2o 8508 df-ixp 8939 df-en 8987 df-fin 8990

This theorem is referenced by: xpsfrn 17614 xpsff1o2 17615

W3C validator