rrx2xpref1o - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > rrx2xpref1o		Structured version Visualization version GIF version

Theorem rrx2xpref1o 48106

Description: There is a bijection between the set of ordered pairs of real numbers (the cartesian product of the real numbers) and the set of points in the two dimensional Euclidean plane (represented as mappings from {1, 2} to the real numbers). (Contributed by AV, 12-Mar-2023.)

**Hypotheses**
Ref	Expression
rrx2xpreen.r	⊢ 𝑅 = (ℝ ↑_m {1, 2})
rrx2xpref1o.1	⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})

**Assertion**
Ref	Expression
rrx2xpref1o	⊢ 𝐹:(ℝ × ℝ)–1-1-onto→𝑅

Distinct variable group: 𝑥,𝑦 Allowed substitution hints: 𝑅(𝑥,𝑦) 𝐹(𝑥,𝑦)

Proof of Theorem rrx2xpref1o Dummy variables 𝑣 𝑢 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrx2xpref1o.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})
2		prex 5438	. . . . 5 ⊢ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} ∈ V
3	1, 2	fnmpoi 8084	. . . 4 ⊢ 𝐹 Fn (ℝ × ℝ)
4		1st2nd2 8042	. . . . . . . . 9 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
5	4	fveq2d 6905	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩))
6		df-ov 7427	. . . . . . . 8 ⊢ ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)) = (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
7	5, 6	eqtr4di 2784	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)))
8		xp1st 8035	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1^st ‘𝑧) ∈ ℝ)
9		xp2nd 8036	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (2^nd ‘𝑧) ∈ ℝ)
10		opeq2 4880	. . . . . . . . . 10 ⊢ (𝑥 = (1^st ‘𝑧) → ⟨1, 𝑥⟩ = ⟨1, (1^st ‘𝑧)⟩)
11	10	preq1d 4748	. . . . . . . . 9 ⊢ (𝑥 = (1^st ‘𝑧) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, (1^st ‘𝑧)⟩, ⟨2, 𝑦⟩})
12		opeq2 4880	. . . . . . . . . 10 ⊢ (𝑦 = (2^nd ‘𝑧) → ⟨2, 𝑦⟩ = ⟨2, (2^nd ‘𝑧)⟩)
13	12	preq2d 4749	. . . . . . . . 9 ⊢ (𝑦 = (2^nd ‘𝑧) → {⟨1, (1^st ‘𝑧)⟩, ⟨2, 𝑦⟩} = {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩})
14		prex 5438	. . . . . . . . 9 ⊢ {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩} ∈ V
15	11, 13, 1, 14	ovmpo 7586	. . . . . . . 8 ⊢ (((1^st ‘𝑧) ∈ ℝ ∧ (2^nd ‘𝑧) ∈ ℝ) → ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)) = {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩})
16	8, 9, 15	syl2anc 582	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)) = {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩})
17	7, 16	eqtrd 2766	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩})
18		eqid 2726	. . . . . . . 8 ⊢ {1, 2} = {1, 2}
19		rrx2xpreen.r	. . . . . . . 8 ⊢ 𝑅 = (ℝ ↑_m {1, 2})
20	18, 19	prelrrx2 48101	. . . . . . 7 ⊢ (((1^st ‘𝑧) ∈ ℝ ∧ (2^nd ‘𝑧) ∈ ℝ) → {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩} ∈ 𝑅)
21	8, 9, 20	syl2anc 582	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩} ∈ 𝑅)
22	17, 21	eqeltrd 2826	. . . . 5 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) ∈ 𝑅)
23	22	rgen 3053	. . . 4 ⊢ ∀𝑧 ∈ (ℝ × ℝ)(𝐹‘𝑧) ∈ 𝑅
24		ffnfv 7133	. . . 4 ⊢ (𝐹:(ℝ × ℝ)⟶𝑅 ↔ (𝐹 Fn (ℝ × ℝ) ∧ ∀𝑧 ∈ (ℝ × ℝ)(𝐹‘𝑧) ∈ 𝑅))
25	3, 23, 24	mpbir2an 709	. . 3 ⊢ 𝐹:(ℝ × ℝ)⟶𝑅
26		opex 5470	. . . . . . . 8 ⊢ ⟨1, (1^st ‘𝑧)⟩ ∈ V
27		opex 5470	. . . . . . . 8 ⊢ ⟨2, (2^nd ‘𝑧)⟩ ∈ V
28		opex 5470	. . . . . . . 8 ⊢ ⟨1, (1^st ‘𝑤)⟩ ∈ V
29		opex 5470	. . . . . . . 8 ⊢ ⟨2, (2^nd ‘𝑤)⟩ ∈ V
30	26, 27, 28, 29	preq12b 4857	. . . . . . 7 ⊢ ({⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩} = {⟨1, (1^st ‘𝑤)⟩, ⟨2, (2^nd ‘𝑤)⟩} ↔ ((⟨1, (1^st ‘𝑧)⟩ = ⟨1, (1^st ‘𝑤)⟩ ∧ ⟨2, (2^nd ‘𝑧)⟩ = ⟨2, (2^nd ‘𝑤)⟩) ∨ (⟨1, (1^st ‘𝑧)⟩ = ⟨2, (2^nd ‘𝑤)⟩ ∧ ⟨2, (2^nd ‘𝑧)⟩ = ⟨1, (1^st ‘𝑤)⟩)))
31		1ex 11260	. . . . . . . . . . . 12 ⊢ 1 ∈ V
32		fvex 6914	. . . . . . . . . . . 12 ⊢ (1^st ‘𝑧) ∈ V
33	31, 32	opth 5482	. . . . . . . . . . 11 ⊢ (⟨1, (1^st ‘𝑧)⟩ = ⟨1, (1^st ‘𝑤)⟩ ↔ (1 = 1 ∧ (1^st ‘𝑧) = (1^st ‘𝑤)))
34	33	simprbi 495	. . . . . . . . . 10 ⊢ (⟨1, (1^st ‘𝑧)⟩ = ⟨1, (1^st ‘𝑤)⟩ → (1^st ‘𝑧) = (1^st ‘𝑤))
35		2ex 12341	. . . . . . . . . . . 12 ⊢ 2 ∈ V
36		fvex 6914	. . . . . . . . . . . 12 ⊢ (2^nd ‘𝑧) ∈ V
37	35, 36	opth 5482	. . . . . . . . . . 11 ⊢ (⟨2, (2^nd ‘𝑧)⟩ = ⟨2, (2^nd ‘𝑤)⟩ ↔ (2 = 2 ∧ (2^nd ‘𝑧) = (2^nd ‘𝑤)))
38	37	simprbi 495	. . . . . . . . . 10 ⊢ (⟨2, (2^nd ‘𝑧)⟩ = ⟨2, (2^nd ‘𝑤)⟩ → (2^nd ‘𝑧) = (2^nd ‘𝑤))
39	34, 38	anim12i 611	. . . . . . . . 9 ⊢ ((⟨1, (1^st ‘𝑧)⟩ = ⟨1, (1^st ‘𝑤)⟩ ∧ ⟨2, (2^nd ‘𝑧)⟩ = ⟨2, (2^nd ‘𝑤)⟩) → ((1^st ‘𝑧) = (1^st ‘𝑤) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑤)))
40	39	a1d 25	. . . . . . . 8 ⊢ ((⟨1, (1^st ‘𝑧)⟩ = ⟨1, (1^st ‘𝑤)⟩ ∧ ⟨2, (2^nd ‘𝑧)⟩ = ⟨2, (2^nd ‘𝑤)⟩) → ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((1^st ‘𝑧) = (1^st ‘𝑤) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑤))))
41	31, 32	opth 5482	. . . . . . . . 9 ⊢ (⟨1, (1^st ‘𝑧)⟩ = ⟨2, (2^nd ‘𝑤)⟩ ↔ (1 = 2 ∧ (1^st ‘𝑧) = (2^nd ‘𝑤)))
42	35, 36	opth 5482	. . . . . . . . 9 ⊢ (⟨2, (2^nd ‘𝑧)⟩ = ⟨1, (1^st ‘𝑤)⟩ ↔ (2 = 1 ∧ (2^nd ‘𝑧) = (1^st ‘𝑤)))
43		1ne2 12472	. . . . . . . . . . 11 ⊢ 1 ≠ 2
44		eqneqall 2941	. . . . . . . . . . 11 ⊢ (1 = 2 → (1 ≠ 2 → ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((1^st ‘𝑧) = (1^st ‘𝑤) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑤)))))
45	43, 44	mpi 20	. . . . . . . . . 10 ⊢ (1 = 2 → ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((1^st ‘𝑧) = (1^st ‘𝑤) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑤))))
46	45	ad2antrr 724	. . . . . . . . 9 ⊢ (((1 = 2 ∧ (1^st ‘𝑧) = (2^nd ‘𝑤)) ∧ (2 = 1 ∧ (2^nd ‘𝑧) = (1^st ‘𝑤))) → ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((1^st ‘𝑧) = (1^st ‘𝑤) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑤))))
47	41, 42, 46	syl2anb 596	. . . . . . . 8 ⊢ ((⟨1, (1^st ‘𝑧)⟩ = ⟨2, (2^nd ‘𝑤)⟩ ∧ ⟨2, (2^nd ‘𝑧)⟩ = ⟨1, (1^st ‘𝑤)⟩) → ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((1^st ‘𝑧) = (1^st ‘𝑤) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑤))))
48	40, 47	jaoi 855	. . . . . . 7 ⊢ (((⟨1, (1^st ‘𝑧)⟩ = ⟨1, (1^st ‘𝑤)⟩ ∧ ⟨2, (2^nd ‘𝑧)⟩ = ⟨2, (2^nd ‘𝑤)⟩) ∨ (⟨1, (1^st ‘𝑧)⟩ = ⟨2, (2^nd ‘𝑤)⟩ ∧ ⟨2, (2^nd ‘𝑧)⟩ = ⟨1, (1^st ‘𝑤)⟩)) → ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((1^st ‘𝑧) = (1^st ‘𝑤) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑤))))
49	30, 48	sylbi 216	. . . . . 6 ⊢ ({⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩} = {⟨1, (1^st ‘𝑤)⟩, ⟨2, (2^nd ‘𝑤)⟩} → ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((1^st ‘𝑧) = (1^st ‘𝑤) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑤))))
50	49	com12 32	. . . . 5 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ({⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩} = {⟨1, (1^st ‘𝑤)⟩, ⟨2, (2^nd ‘𝑤)⟩} → ((1^st ‘𝑧) = (1^st ‘𝑤) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑤))))
51		1st2nd2 8042	. . . . . . . . 9 ⊢ (𝑤 ∈ (ℝ × ℝ) → 𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
52	51	fveq2d 6905	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → (𝐹‘𝑤) = (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩))
53		df-ov 7427	. . . . . . . 8 ⊢ ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)) = (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
54	52, 53	eqtr4di 2784	. . . . . . 7 ⊢ (𝑤 ∈ (ℝ × ℝ) → (𝐹‘𝑤) = ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)))
55		xp1st 8035	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → (1^st ‘𝑤) ∈ ℝ)
56		xp2nd 8036	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → (2^nd ‘𝑤) ∈ ℝ)
57		opeq2 4880	. . . . . . . . . 10 ⊢ (𝑥 = (1^st ‘𝑤) → ⟨1, 𝑥⟩ = ⟨1, (1^st ‘𝑤)⟩)
58	57	preq1d 4748	. . . . . . . . 9 ⊢ (𝑥 = (1^st ‘𝑤) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, (1^st ‘𝑤)⟩, ⟨2, 𝑦⟩})
59		opeq2 4880	. . . . . . . . . 10 ⊢ (𝑦 = (2^nd ‘𝑤) → ⟨2, 𝑦⟩ = ⟨2, (2^nd ‘𝑤)⟩)
60	59	preq2d 4749	. . . . . . . . 9 ⊢ (𝑦 = (2^nd ‘𝑤) → {⟨1, (1^st ‘𝑤)⟩, ⟨2, 𝑦⟩} = {⟨1, (1^st ‘𝑤)⟩, ⟨2, (2^nd ‘𝑤)⟩})
61		prex 5438	. . . . . . . . 9 ⊢ {⟨1, (1^st ‘𝑤)⟩, ⟨2, (2^nd ‘𝑤)⟩} ∈ V
62	58, 60, 1, 61	ovmpo 7586	. . . . . . . 8 ⊢ (((1^st ‘𝑤) ∈ ℝ ∧ (2^nd ‘𝑤) ∈ ℝ) → ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)) = {⟨1, (1^st ‘𝑤)⟩, ⟨2, (2^nd ‘𝑤)⟩})
63	55, 56, 62	syl2anc 582	. . . . . . 7 ⊢ (𝑤 ∈ (ℝ × ℝ) → ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)) = {⟨1, (1^st ‘𝑤)⟩, ⟨2, (2^nd ‘𝑤)⟩})
64	54, 63	eqtrd 2766	. . . . . 6 ⊢ (𝑤 ∈ (ℝ × ℝ) → (𝐹‘𝑤) = {⟨1, (1^st ‘𝑤)⟩, ⟨2, (2^nd ‘𝑤)⟩})
65	17, 64	eqeqan12d 2740	. . . . 5 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩} = {⟨1, (1^st ‘𝑤)⟩, ⟨2, (2^nd ‘𝑤)⟩}))
66	4, 51	eqeqan12d 2740	. . . . . 6 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (𝑧 = 𝑤 ↔ ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩))
67	32, 36	opth 5482	. . . . . 6 ⊢ (⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩ ↔ ((1^st ‘𝑧) = (1^st ‘𝑤) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑤)))
68	66, 67	bitrdi 286	. . . . 5 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (𝑧 = 𝑤 ↔ ((1^st ‘𝑧) = (1^st ‘𝑤) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑤))))
69	50, 65, 68	3imtr4d 293	. . . 4 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
70	69	rgen2 3188	. . 3 ⊢ ∀𝑧 ∈ (ℝ × ℝ)∀𝑤 ∈ (ℝ × ℝ)((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)
71		dff13 7270	. . 3 ⊢ (𝐹:(ℝ × ℝ)–1-1→𝑅 ↔ (𝐹:(ℝ × ℝ)⟶𝑅 ∧ ∀𝑧 ∈ (ℝ × ℝ)∀𝑤 ∈ (ℝ × ℝ)((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)))
72	25, 70, 71	mpbir2an 709	. 2 ⊢ 𝐹:(ℝ × ℝ)–1-1→𝑅
73	19	eleq2i 2818	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑅 ↔ 𝑤 ∈ (ℝ ↑_m {1, 2}))
74		reex 11249	. . . . . . . . 9 ⊢ ℝ ∈ V
75		prex 5438	. . . . . . . . 9 ⊢ {1, 2} ∈ V
76	74, 75	elmap 8900	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ ↑_m {1, 2}) ↔ 𝑤:{1, 2}⟶ℝ)
77		1re 11264	. . . . . . . . 9 ⊢ 1 ∈ ℝ
78		2re 12338	. . . . . . . . 9 ⊢ 2 ∈ ℝ
79		fpr2g 7228	. . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ 2 ∈ ℝ) → (𝑤:{1, 2}⟶ℝ ↔ ((𝑤‘1) ∈ ℝ ∧ (𝑤‘2) ∈ ℝ ∧ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩})))
80	77, 78, 79	mp2an 690	. . . . . . . 8 ⊢ (𝑤:{1, 2}⟶ℝ ↔ ((𝑤‘1) ∈ ℝ ∧ (𝑤‘2) ∈ ℝ ∧ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩}))
81	73, 76, 80	3bitri 296	. . . . . . 7 ⊢ (𝑤 ∈ 𝑅 ↔ ((𝑤‘1) ∈ ℝ ∧ (𝑤‘2) ∈ ℝ ∧ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩}))
82		opeq2 4880	. . . . . . . . . 10 ⊢ (𝑢 = (𝑤‘1) → ⟨1, 𝑢⟩ = ⟨1, (𝑤‘1)⟩)
83	82	preq1d 4748	. . . . . . . . 9 ⊢ (𝑢 = (𝑤‘1) → {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩})
84	83	eqeq2d 2737	. . . . . . . 8 ⊢ (𝑢 = (𝑤‘1) → (𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} ↔ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩}))
85		opeq2 4880	. . . . . . . . . 10 ⊢ (𝑣 = (𝑤‘2) → ⟨2, 𝑣⟩ = ⟨2, (𝑤‘2)⟩)
86	85	preq2d 4749	. . . . . . . . 9 ⊢ (𝑣 = (𝑤‘2) → {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩} = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩})
87	86	eqeq2d 2737	. . . . . . . 8 ⊢ (𝑣 = (𝑤‘2) → (𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩} ↔ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩}))
88	84, 87	rspc2ev 3621	. . . . . . 7 ⊢ (((𝑤‘1) ∈ ℝ ∧ (𝑤‘2) ∈ ℝ ∧ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩}) → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
89	81, 88	sylbi 216	. . . . . 6 ⊢ (𝑤 ∈ 𝑅 → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
90		opeq2 4880	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → ⟨1, 𝑥⟩ = ⟨1, 𝑢⟩)
91	90	preq1d 4748	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑢⟩, ⟨2, 𝑦⟩})
92		opeq2 4880	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → ⟨2, 𝑦⟩ = ⟨2, 𝑣⟩)
93	92	preq2d 4749	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → {⟨1, 𝑢⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
94		prex 5438	. . . . . . . . 9 ⊢ {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} ∈ V
95	91, 93, 1, 94	ovmpo 7586	. . . . . . . 8 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢𝐹𝑣) = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
96	95	eqeq2d 2737	. . . . . . 7 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑤 = (𝑢𝐹𝑣) ↔ 𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩}))
97	96	2rexbiia 3206	. . . . . 6 ⊢ (∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
98	89, 97	sylibr 233	. . . . 5 ⊢ (𝑤 ∈ 𝑅 → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
99		fveq2 6901	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝑧) = (𝐹‘⟨𝑢, 𝑣⟩))
100		df-ov 7427	. . . . . . . 8 ⊢ (𝑢𝐹𝑣) = (𝐹‘⟨𝑢, 𝑣⟩)
101	99, 100	eqtr4di 2784	. . . . . . 7 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝑧) = (𝑢𝐹𝑣))
102	101	eqeq2d 2737	. . . . . 6 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝑤 = (𝐹‘𝑧) ↔ 𝑤 = (𝑢𝐹𝑣)))
103	102	rexxp 5849	. . . . 5 ⊢ (∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
104	98, 103	sylibr 233	. . . 4 ⊢ (𝑤 ∈ 𝑅 → ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧))
105	104	rgen 3053	. . 3 ⊢ ∀𝑤 ∈ 𝑅 ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧)
106		dffo3 7116	. . 3 ⊢ (𝐹:(ℝ × ℝ)–onto→𝑅 ↔ (𝐹:(ℝ × ℝ)⟶𝑅 ∧ ∀𝑤 ∈ 𝑅 ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧)))
107	25, 105, 106	mpbir2an 709	. 2 ⊢ 𝐹:(ℝ × ℝ)–onto→𝑅
108		df-f1o 6561	. 2 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→𝑅 ↔ (𝐹:(ℝ × ℝ)–1-1→𝑅 ∧ 𝐹:(ℝ × ℝ)–onto→𝑅))
109	72, 107, 108	mpbir2an 709	1 ⊢ 𝐹:(ℝ × ℝ)–1-1-onto→𝑅

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 {cpr 4635 ⟨cop 4639 × cxp 5680 Fn wfn 6549 ⟶wf 6550 –1-1→wf1 6551 –onto→wfo 6552 –1-1-onto→wf1o 6553 ‘cfv 6554 (class class class)co 7424 ∈ cmpo 7426 1^st c1st 8001 2^nd c2nd 8002 ↑_m cmap 8855 ℝcr 11157 1c1 11159 2c2 12319

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-2 12327

This theorem is referenced by: rrx2xpreen 48107 rrx2plordisom 48111

W3C validator