rrx2xpref1o - Mathbox for Alexander van der Vekens

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Theorem rrx2xpref1o 46064

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 5355	. . . . 5 ⊢ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} ∈ V
3	1, 2	fnmpoi 7910	. . . 4 ⊢ 𝐹 Fn (ℝ × ℝ)
4		1st2nd2 7870	. . . . . . . . 9 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
5	4	fveq2d 6778	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩))
6		df-ov 7278	. . . . . . . 8 ⊢ ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)) = (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
7	5, 6	eqtr4di 2796	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)))
8		xp1st 7863	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1^st ‘𝑧) ∈ ℝ)
9		xp2nd 7864	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (2^nd ‘𝑧) ∈ ℝ)
10		opeq2 4805	. . . . . . . . . 10 ⊢ (𝑥 = (1^st ‘𝑧) → ⟨1, 𝑥⟩ = ⟨1, (1^st ‘𝑧)⟩)
11	10	preq1d 4675	. . . . . . . . 9 ⊢ (𝑥 = (1^st ‘𝑧) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, (1^st ‘𝑧)⟩, ⟨2, 𝑦⟩})
12		opeq2 4805	. . . . . . . . . 10 ⊢ (𝑦 = (2^nd ‘𝑧) → ⟨2, 𝑦⟩ = ⟨2, (2^nd ‘𝑧)⟩)
13	12	preq2d 4676	. . . . . . . . 9 ⊢ (𝑦 = (2^nd ‘𝑧) → {⟨1, (1^st ‘𝑧)⟩, ⟨2, 𝑦⟩} = {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩})
14		prex 5355	. . . . . . . . 9 ⊢ {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩} ∈ V
15	11, 13, 1, 14	ovmpo 7433	. . . . . . . 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 584	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)) = {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩})
17	7, 16	eqtrd 2778	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩})
18		eqid 2738	. . . . . . . 8 ⊢ {1, 2} = {1, 2}
19		rrx2xpreen.r	. . . . . . . 8 ⊢ 𝑅 = (ℝ ↑_m {1, 2})
20	18, 19	prelrrx2 46059	. . . . . . 7 ⊢ (((1^st ‘𝑧) ∈ ℝ ∧ (2^nd ‘𝑧) ∈ ℝ) → {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩} ∈ 𝑅)
21	8, 9, 20	syl2anc 584	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩} ∈ 𝑅)
22	17, 21	eqeltrd 2839	. . . . 5 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) ∈ 𝑅)
23	22	rgen 3074	. . . 4 ⊢ ∀𝑧 ∈ (ℝ × ℝ)(𝐹‘𝑧) ∈ 𝑅
24		ffnfv 6992	. . . 4 ⊢ (𝐹:(ℝ × ℝ)⟶𝑅 ↔ (𝐹 Fn (ℝ × ℝ) ∧ ∀𝑧 ∈ (ℝ × ℝ)(𝐹‘𝑧) ∈ 𝑅))
25	3, 23, 24	mpbir2an 708	. . 3 ⊢ 𝐹:(ℝ × ℝ)⟶𝑅
26		opex 5379	. . . . . . . 8 ⊢ ⟨1, (1^st ‘𝑧)⟩ ∈ V
27		opex 5379	. . . . . . . 8 ⊢ ⟨2, (2^nd ‘𝑧)⟩ ∈ V
28		opex 5379	. . . . . . . 8 ⊢ ⟨1, (1^st ‘𝑤)⟩ ∈ V
29		opex 5379	. . . . . . . 8 ⊢ ⟨2, (2^nd ‘𝑤)⟩ ∈ V
30	26, 27, 28, 29	preq12b 4781	. . . . . . 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 10971	. . . . . . . . . . . 12 ⊢ 1 ∈ V
32		fvex 6787	. . . . . . . . . . . 12 ⊢ (1^st ‘𝑧) ∈ V
33	31, 32	opth 5391	. . . . . . . . . . 11 ⊢ (⟨1, (1^st ‘𝑧)⟩ = ⟨1, (1^st ‘𝑤)⟩ ↔ (1 = 1 ∧ (1^st ‘𝑧) = (1^st ‘𝑤)))
34	33	simprbi 497	. . . . . . . . . 10 ⊢ (⟨1, (1^st ‘𝑧)⟩ = ⟨1, (1^st ‘𝑤)⟩ → (1^st ‘𝑧) = (1^st ‘𝑤))
35		2ex 12050	. . . . . . . . . . . 12 ⊢ 2 ∈ V
36		fvex 6787	. . . . . . . . . . . 12 ⊢ (2^nd ‘𝑧) ∈ V
37	35, 36	opth 5391	. . . . . . . . . . 11 ⊢ (⟨2, (2^nd ‘𝑧)⟩ = ⟨2, (2^nd ‘𝑤)⟩ ↔ (2 = 2 ∧ (2^nd ‘𝑧) = (2^nd ‘𝑤)))
38	37	simprbi 497	. . . . . . . . . 10 ⊢ (⟨2, (2^nd ‘𝑧)⟩ = ⟨2, (2^nd ‘𝑤)⟩ → (2^nd ‘𝑧) = (2^nd ‘𝑤))
39	34, 38	anim12i 613	. . . . . . . . 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 5391	. . . . . . . . 9 ⊢ (⟨1, (1^st ‘𝑧)⟩ = ⟨2, (2^nd ‘𝑤)⟩ ↔ (1 = 2 ∧ (1^st ‘𝑧) = (2^nd ‘𝑤)))
42	35, 36	opth 5391	. . . . . . . . 9 ⊢ (⟨2, (2^nd ‘𝑧)⟩ = ⟨1, (1^st ‘𝑤)⟩ ↔ (2 = 1 ∧ (2^nd ‘𝑧) = (1^st ‘𝑤)))
43		1ne2 12181	. . . . . . . . . . 11 ⊢ 1 ≠ 2
44		eqneqall 2954	. . . . . . . . . . 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 723	. . . . . . . . 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 598	. . . . . . . 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 854	. . . . . . 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 7870	. . . . . . . . 9 ⊢ (𝑤 ∈ (ℝ × ℝ) → 𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
52	51	fveq2d 6778	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → (𝐹‘𝑤) = (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩))
53		df-ov 7278	. . . . . . . 8 ⊢ ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)) = (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
54	52, 53	eqtr4di 2796	. . . . . . 7 ⊢ (𝑤 ∈ (ℝ × ℝ) → (𝐹‘𝑤) = ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)))
55		xp1st 7863	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → (1^st ‘𝑤) ∈ ℝ)
56		xp2nd 7864	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → (2^nd ‘𝑤) ∈ ℝ)
57		opeq2 4805	. . . . . . . . . 10 ⊢ (𝑥 = (1^st ‘𝑤) → ⟨1, 𝑥⟩ = ⟨1, (1^st ‘𝑤)⟩)
58	57	preq1d 4675	. . . . . . . . 9 ⊢ (𝑥 = (1^st ‘𝑤) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, (1^st ‘𝑤)⟩, ⟨2, 𝑦⟩})
59		opeq2 4805	. . . . . . . . . 10 ⊢ (𝑦 = (2^nd ‘𝑤) → ⟨2, 𝑦⟩ = ⟨2, (2^nd ‘𝑤)⟩)
60	59	preq2d 4676	. . . . . . . . 9 ⊢ (𝑦 = (2^nd ‘𝑤) → {⟨1, (1^st ‘𝑤)⟩, ⟨2, 𝑦⟩} = {⟨1, (1^st ‘𝑤)⟩, ⟨2, (2^nd ‘𝑤)⟩})
61		prex 5355	. . . . . . . . 9 ⊢ {⟨1, (1^st ‘𝑤)⟩, ⟨2, (2^nd ‘𝑤)⟩} ∈ V
62	58, 60, 1, 61	ovmpo 7433	. . . . . . . 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 584	. . . . . . 7 ⊢ (𝑤 ∈ (ℝ × ℝ) → ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)) = {⟨1, (1^st ‘𝑤)⟩, ⟨2, (2^nd ‘𝑤)⟩})
64	54, 63	eqtrd 2778	. . . . . 6 ⊢ (𝑤 ∈ (ℝ × ℝ) → (𝐹‘𝑤) = {⟨1, (1^st ‘𝑤)⟩, ⟨2, (2^nd ‘𝑤)⟩})
65	17, 64	eqeqan12d 2752	. . . . 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 2752	. . . . . 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 5391	. . . . . 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 287	. . . . 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 294	. . . 4 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
70	69	rgen2 3120	. . 3 ⊢ ∀𝑧 ∈ (ℝ × ℝ)∀𝑤 ∈ (ℝ × ℝ)((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)
71		dff13 7128	. . 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 708	. 2 ⊢ 𝐹:(ℝ × ℝ)–1-1→𝑅
73	19	eleq2i 2830	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑅 ↔ 𝑤 ∈ (ℝ ↑_m {1, 2}))
74		reex 10962	. . . . . . . . 9 ⊢ ℝ ∈ V
75		prex 5355	. . . . . . . . 9 ⊢ {1, 2} ∈ V
76	74, 75	elmap 8659	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ ↑_m {1, 2}) ↔ 𝑤:{1, 2}⟶ℝ)
77		1re 10975	. . . . . . . . 9 ⊢ 1 ∈ ℝ
78		2re 12047	. . . . . . . . 9 ⊢ 2 ∈ ℝ
79		fpr2g 7087	. . . . . . . . 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 689	. . . . . . . 8 ⊢ (𝑤:{1, 2}⟶ℝ ↔ ((𝑤‘1) ∈ ℝ ∧ (𝑤‘2) ∈ ℝ ∧ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩}))
81	73, 76, 80	3bitri 297	. . . . . . 7 ⊢ (𝑤 ∈ 𝑅 ↔ ((𝑤‘1) ∈ ℝ ∧ (𝑤‘2) ∈ ℝ ∧ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩}))
82		opeq2 4805	. . . . . . . . . 10 ⊢ (𝑢 = (𝑤‘1) → ⟨1, 𝑢⟩ = ⟨1, (𝑤‘1)⟩)
83	82	preq1d 4675	. . . . . . . . 9 ⊢ (𝑢 = (𝑤‘1) → {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩})
84	83	eqeq2d 2749	. . . . . . . 8 ⊢ (𝑢 = (𝑤‘1) → (𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} ↔ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩}))
85		opeq2 4805	. . . . . . . . . 10 ⊢ (𝑣 = (𝑤‘2) → ⟨2, 𝑣⟩ = ⟨2, (𝑤‘2)⟩)
86	85	preq2d 4676	. . . . . . . . 9 ⊢ (𝑣 = (𝑤‘2) → {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩} = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩})
87	86	eqeq2d 2749	. . . . . . . 8 ⊢ (𝑣 = (𝑤‘2) → (𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩} ↔ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩}))
88	84, 87	rspc2ev 3572	. . . . . . 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 4805	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → ⟨1, 𝑥⟩ = ⟨1, 𝑢⟩)
91	90	preq1d 4675	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑢⟩, ⟨2, 𝑦⟩})
92		opeq2 4805	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → ⟨2, 𝑦⟩ = ⟨2, 𝑣⟩)
93	92	preq2d 4676	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → {⟨1, 𝑢⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
94		prex 5355	. . . . . . . . 9 ⊢ {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} ∈ V
95	91, 93, 1, 94	ovmpo 7433	. . . . . . . 8 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢𝐹𝑣) = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
96	95	eqeq2d 2749	. . . . . . 7 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑤 = (𝑢𝐹𝑣) ↔ 𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩}))
97	96	2rexbiia 3227	. . . . . 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 6774	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝑧) = (𝐹‘⟨𝑢, 𝑣⟩))
100		df-ov 7278	. . . . . . . 8 ⊢ (𝑢𝐹𝑣) = (𝐹‘⟨𝑢, 𝑣⟩)
101	99, 100	eqtr4di 2796	. . . . . . 7 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝑧) = (𝑢𝐹𝑣))
102	101	eqeq2d 2749	. . . . . 6 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝑤 = (𝐹‘𝑧) ↔ 𝑤 = (𝑢𝐹𝑣)))
103	102	rexxp 5751	. . . . 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 3074	. . 3 ⊢ ∀𝑤 ∈ 𝑅 ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧)
106		dffo3 6978	. . 3 ⊢ (𝐹:(ℝ × ℝ)–onto→𝑅 ↔ (𝐹:(ℝ × ℝ)⟶𝑅 ∧ ∀𝑤 ∈ 𝑅 ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧)))
107	25, 105, 106	mpbir2an 708	. 2 ⊢ 𝐹:(ℝ × ℝ)–onto→𝑅
108		df-f1o 6440	. 2 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→𝑅 ↔ (𝐹:(ℝ × ℝ)–1-1→𝑅 ∧ 𝐹:(ℝ × ℝ)–onto→𝑅))
109	72, 107, 108	mpbir2an 708	1 ⊢ 𝐹:(ℝ × ℝ)–1-1-onto→𝑅

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3065 {cpr 4563 ⟨cop 4567 × cxp 5587 Fn wfn 6428 ⟶wf 6429 –1-1→wf1 6430 –onto→wfo 6431 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 ∈ cmpo 7277 1^st c1st 7829 2^nd c2nd 7830 ↑_m cmap 8615 ℝcr 10870 1c1 10872 2c2 12028

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-2 12036

This theorem is referenced by: rrx2xpreen 46065 rrx2plordisom 46069

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