xpsff1o - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > xpsff1o		GIF version

Theorem xpsff1o 12787

Description: The function appearing in xpsval 12790 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 12784	. . . . . 6 ⊢ ({⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩} ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
2	1	biimpri 133	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → {⟨∅, 𝑥⟩, ⟨1_o, 𝑦⟩} ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵))
3	2	rgen2 2573	. . . 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 6216	. . . 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 145	. . 3 ⊢ 𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)
7		1st2nd2 6190	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
8	7	fveq2d 5531	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘𝑧) = (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩))
9		df-ov 5891	. . . . . . . 8 ⊢ ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)) = (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
10		xp1st 6180	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (1^st ‘𝑧) ∈ 𝐴)
11		xp2nd 6181	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (2^nd ‘𝑧) ∈ 𝐵)
12	4	xpsfval 12786	. . . . . . . . 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 411	. . . . . . . 8 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)) = {⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩})
14	9, 13	eqtr3id 2234	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩) = {⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩})
15	8, 14	eqtrd 2220	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘𝑧) = {⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩})
16		1st2nd2 6190	. . . . . . . 8 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → 𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
17	16	fveq2d 5531	. . . . . . 7 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘𝑤) = (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩))
18		df-ov 5891	. . . . . . . 8 ⊢ ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)) = (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
19		xp1st 6180	. . . . . . . . 9 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (1^st ‘𝑤) ∈ 𝐴)
20		xp2nd 6181	. . . . . . . . 9 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (2^nd ‘𝑤) ∈ 𝐵)
21	4	xpsfval 12786	. . . . . . . . 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 411	. . . . . . . 8 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)) = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩})
23	18, 22	eqtr3id 2234	. . . . . . 7 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩) = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩})
24	17, 23	eqtrd 2220	. . . . . 6 ⊢ (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘𝑤) = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩})
25	15, 24	eqeqan12d 2203	. . . . 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 5526	. . . . . . . 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		1stexg 6182	. . . . . . . . . 10 ⊢ (𝑧 ∈ V → (1^st ‘𝑧) ∈ V)
28	27	elv 2753	. . . . . . . . 9 ⊢ (1^st ‘𝑧) ∈ V
29		fvpr0o 12779	. . . . . . . . 9 ⊢ ((1^st ‘𝑧) ∈ V → ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩}‘∅) = (1^st ‘𝑧))
30	28, 29	ax-mp 5	. . . . . . . 8 ⊢ ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩}‘∅) = (1^st ‘𝑧)
31		1stexg 6182	. . . . . . . . . 10 ⊢ (𝑤 ∈ V → (1^st ‘𝑤) ∈ V)
32	31	elv 2753	. . . . . . . . 9 ⊢ (1^st ‘𝑤) ∈ V
33		fvpr0o 12779	. . . . . . . . 9 ⊢ ((1^st ‘𝑤) ∈ V → ({⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩}‘∅) = (1^st ‘𝑤))
34	32, 33	ax-mp 5	. . . . . . . 8 ⊢ ({⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩}‘∅) = (1^st ‘𝑤)
35	26, 30, 34	3eqtr3g 2243	. . . . . . 7 ⊢ ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩} = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩} → (1^st ‘𝑧) = (1^st ‘𝑤))
36		fveq1 5526	. . . . . . . 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))
37		2ndexg 6183	. . . . . . . . . 10 ⊢ (𝑧 ∈ V → (2^nd ‘𝑧) ∈ V)
38	37	elv 2753	. . . . . . . . 9 ⊢ (2^nd ‘𝑧) ∈ V
39		fvpr1o 12780	. . . . . . . . 9 ⊢ ((2^nd ‘𝑧) ∈ V → ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩}‘1_o) = (2^nd ‘𝑧))
40	38, 39	ax-mp 5	. . . . . . . 8 ⊢ ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩}‘1_o) = (2^nd ‘𝑧)
41		2ndexg 6183	. . . . . . . . . 10 ⊢ (𝑤 ∈ V → (2^nd ‘𝑤) ∈ V)
42	41	elv 2753	. . . . . . . . 9 ⊢ (2^nd ‘𝑤) ∈ V
43		fvpr1o 12780	. . . . . . . . 9 ⊢ ((2^nd ‘𝑤) ∈ V → ({⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩}‘1_o) = (2^nd ‘𝑤))
44	42, 43	ax-mp 5	. . . . . . . 8 ⊢ ({⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩}‘1_o) = (2^nd ‘𝑤)
45	36, 40, 44	3eqtr3g 2243	. . . . . . 7 ⊢ ({⟨∅, (1^st ‘𝑧)⟩, ⟨1_o, (2^nd ‘𝑧)⟩} = {⟨∅, (1^st ‘𝑤)⟩, ⟨1_o, (2^nd ‘𝑤)⟩} → (2^nd ‘𝑧) = (2^nd ‘𝑤))
46	35, 45	opeq12d 3798	. . . . . 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)⟩)
47	7, 16	eqeqan12d 2203	. . . . . 6 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → (𝑧 = 𝑤 ↔ ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩))
48	46, 47	imbitrrid 156	. . . . 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))
49	25, 48	sylbid 150	. . . 4 ⊢ ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
50	49	rgen2 2573	. . 3 ⊢ ∀𝑧 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)
51		dff13 5782	. . 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)))
52	6, 50, 51	mpbir2an 943	. 2 ⊢ 𝐹:(𝐴 × 𝐵)–1-1→X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)
53		xpsfrnel 12782	. . . . . 6 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑧 Fn 2_o ∧ (𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1_o) ∈ 𝐵))
54	53	simp2bi 1014	. . . . 5 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘∅) ∈ 𝐴)
55	53	simp3bi 1015	. . . . 5 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘1_o) ∈ 𝐵)
56	4	xpsfval 12786	. . . . . . 7 ⊢ (((𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1_o) ∈ 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1_o)) = {⟨∅, (𝑧‘∅)⟩, ⟨1_o, (𝑧‘1_o)⟩})
57	54, 55, 56	syl2anc 411	. . . . . 6 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1_o)) = {⟨∅, (𝑧‘∅)⟩, ⟨1_o, (𝑧‘1_o)⟩})
58		ixpfn 6718	. . . . . . 7 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 Fn 2_o)
59		xpsfeq 12783	. . . . . . 7 ⊢ (𝑧 Fn 2_o → {⟨∅, (𝑧‘∅)⟩, ⟨1_o, (𝑧‘1_o)⟩} = 𝑧)
60	58, 59	syl 14	. . . . . 6 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → {⟨∅, (𝑧‘∅)⟩, ⟨1_o, (𝑧‘1_o)⟩} = 𝑧)
61	57, 60	eqtr2d 2221	. . . . 5 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 = ((𝑧‘∅)𝐹(𝑧‘1_o)))
62		rspceov 5930	. . . . 5 ⊢ (((𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1_o) ∈ 𝐵 ∧ 𝑧 = ((𝑧‘∅)𝐹(𝑧‘1_o))) → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎𝐹𝑏))
63	54, 55, 61, 62	syl3anc 1248	. . . 4 ⊢ (𝑧 ∈ X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵) → ∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎𝐹𝑏))
64	63	rgen 2540	. . 3 ⊢ ∀𝑧 ∈ X 𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)∃𝑎 ∈ 𝐴 ∃𝑏 ∈ 𝐵 𝑧 = (𝑎𝐹𝑏)
65		foov 6035	. . 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)))
66	6, 64, 65	mpbir2an 943	. 2 ⊢ 𝐹:(𝐴 × 𝐵)–onto→X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)
67		df-f1o 5235	. 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)))
68	52, 66, 67	mpbir2an 943	1 ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→X𝑘 ∈ 2_o if(𝑘 = ∅, 𝐴, 𝐵)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1363 ∈ wcel 2158 ∀wral 2465 ∃wrex 2466 Vcvv 2749 ∅c0 3434 ifcif 3546 {cpr 3605 ⟨cop 3607 × cxp 4636 Fn wfn 5223 ⟶wf 5224 –1-1→wf1 5225 –onto→wfo 5226 –1-1-onto→wf1o 5227 ‘cfv 5228 (class class class)co 5888 ∈ cmpo 5890 1^st c1st 6153 2^nd c2nd 6154 1_oc1o 6424 2_oc2o 6425 Xcixp 6712

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-iord 4378 df-on 4380 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6155 df-2nd 6156 df-1o 6431 df-2o 6432 df-er 6549 df-ixp 6713 df-en 6755 df-fin 6757

This theorem is referenced by: xpsfrn 12788 xpsff1o2 12789

W3C validator