rrx2xpref1o - Mathbox for Alexander van der Vekens

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Theorem rrx2xpref1o 49829

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 5396	. . . . 5 ⊢ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} ∈ V
3	1, 2	fnmpoi 8081	. . . 4 ⊢ 𝐹 Fn (ℝ × ℝ)
4		1st2nd2 8040	. . . . . . . . 9 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
5	4	fveq2d 6889	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩))
6		df-ov 7423	. . . . . . . 8 ⊢ ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)) = (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
7	5, 6	eqtr4di 2814	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)))
8		xp1st 8033	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1^st ‘𝑧) ∈ ℝ)
9		xp2nd 8034	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (2^nd ‘𝑧) ∈ ℝ)
10		opeq2 4834	. . . . . . . . . 10 ⊢ (𝑥 = (1^st ‘𝑧) → ⟨1, 𝑥⟩ = ⟨1, (1^st ‘𝑧)⟩)
11	10	preq1d 4700	. . . . . . . . 9 ⊢ (𝑥 = (1^st ‘𝑧) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, (1^st ‘𝑧)⟩, ⟨2, 𝑦⟩})
12		opeq2 4834	. . . . . . . . . 10 ⊢ (𝑦 = (2^nd ‘𝑧) → ⟨2, 𝑦⟩ = ⟨2, (2^nd ‘𝑧)⟩)
13	12	preq2d 4701	. . . . . . . . 9 ⊢ (𝑦 = (2^nd ‘𝑧) → {⟨1, (1^st ‘𝑧)⟩, ⟨2, 𝑦⟩} = {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩})
14		prex 5396	. . . . . . . . 9 ⊢ {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩} ∈ V
15	11, 13, 1, 14	ovmpo 7580	. . . . . . . 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 596	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)) = {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩})
17	7, 16	eqtrd 2796	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩})
18		eqid 2761	. . . . . . . 8 ⊢ {1, 2} = {1, 2}
19		rrx2xpreen.r	. . . . . . . 8 ⊢ 𝑅 = (ℝ ↑_m {1, 2})
20	18, 19	prelrrx2 49824	. . . . . . 7 ⊢ (((1^st ‘𝑧) ∈ ℝ ∧ (2^nd ‘𝑧) ∈ ℝ) → {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩} ∈ 𝑅)
21	8, 9, 20	syl2anc 596	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩} ∈ 𝑅)
22	17, 21	eqeltrd 2861	. . . . 5 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) ∈ 𝑅)
23	22	rgen 3079	. . . 4 ⊢ ∀𝑧 ∈ (ℝ × ℝ)(𝐹‘𝑧) ∈ 𝑅
24		ffnfv 7119	. . . 4 ⊢ (𝐹:(ℝ × ℝ)⟶𝑅 ↔ (𝐹 Fn (ℝ × ℝ) ∧ ∀𝑧 ∈ (ℝ × ℝ)(𝐹‘𝑧) ∈ 𝑅))
25	3, 23, 24	mpbir2an 724	. . 3 ⊢ 𝐹:(ℝ × ℝ)⟶𝑅
26		opex 5432	. . . . . . . 8 ⊢ ⟨1, (1^st ‘𝑧)⟩ ∈ V
27		opex 5432	. . . . . . . 8 ⊢ ⟨2, (2^nd ‘𝑧)⟩ ∈ V
28		opex 5432	. . . . . . . 8 ⊢ ⟨1, (1^st ‘𝑤)⟩ ∈ V
29		opex 5432	. . . . . . . 8 ⊢ ⟨2, (2^nd ‘𝑤)⟩ ∈ V
30	26, 27, 28, 29	preq12b 4810	. . . . . . 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 11303	. . . . . . . . . . . 12 ⊢ 1 ∈ V
32		fvex 6898	. . . . . . . . . . . 12 ⊢ (1^st ‘𝑧) ∈ V
33	31, 32	opth 5445	. . . . . . . . . . 11 ⊢ (⟨1, (1^st ‘𝑧)⟩ = ⟨1, (1^st ‘𝑤)⟩ ↔ (1 = 1 ∧ (1^st ‘𝑧) = (1^st ‘𝑤)))
34	33	simprbi 503	. . . . . . . . . 10 ⊢ (⟨1, (1^st ‘𝑧)⟩ = ⟨1, (1^st ‘𝑤)⟩ → (1^st ‘𝑧) = (1^st ‘𝑤))
35		2ex 12420	. . . . . . . . . . . 12 ⊢ 2 ∈ V
36		fvex 6898	. . . . . . . . . . . 12 ⊢ (2^nd ‘𝑧) ∈ V
37	35, 36	opth 5445	. . . . . . . . . . 11 ⊢ (⟨2, (2^nd ‘𝑧)⟩ = ⟨2, (2^nd ‘𝑤)⟩ ↔ (2 = 2 ∧ (2^nd ‘𝑧) = (2^nd ‘𝑤)))
38	37	simprbi 503	. . . . . . . . . 10 ⊢ (⟨2, (2^nd ‘𝑧)⟩ = ⟨2, (2^nd ‘𝑤)⟩ → (2^nd ‘𝑧) = (2^nd ‘𝑤))
39	34, 38	anim12i 625	. . . . . . . . 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 26	. . . . . . . 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 5445	. . . . . . . . 9 ⊢ (⟨1, (1^st ‘𝑧)⟩ = ⟨2, (2^nd ‘𝑤)⟩ ↔ (1 = 2 ∧ (1^st ‘𝑧) = (2^nd ‘𝑤)))
42	35, 36	opth 5445	. . . . . . . . 9 ⊢ (⟨2, (2^nd ‘𝑧)⟩ = ⟨1, (1^st ‘𝑤)⟩ ↔ (2 = 1 ∧ (2^nd ‘𝑧) = (1^st ‘𝑤)))
43		1ne2 12553	. . . . . . . . . . 11 ⊢ 1 ≠ 2
44		eqneqall 2967	. . . . . . . . . . 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 21	. . . . . . . . . 10 ⊢ (1 = 2 → ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((1^st ‘𝑧) = (1^st ‘𝑤) ∧ (2^nd ‘𝑧) = (2^nd ‘𝑤))))
46	45	ad2antrr 739	. . . . . . . . 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 610	. . . . . . . 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 871	. . . . . . 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 220	. . . . . 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 33	. . . . 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 8040	. . . . . . . . 9 ⊢ (𝑤 ∈ (ℝ × ℝ) → 𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
52	51	fveq2d 6889	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → (𝐹‘𝑤) = (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩))
53		df-ov 7423	. . . . . . . 8 ⊢ ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)) = (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
54	52, 53	eqtr4di 2814	. . . . . . 7 ⊢ (𝑤 ∈ (ℝ × ℝ) → (𝐹‘𝑤) = ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)))
55		xp1st 8033	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → (1^st ‘𝑤) ∈ ℝ)
56		xp2nd 8034	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → (2^nd ‘𝑤) ∈ ℝ)
57		opeq2 4834	. . . . . . . . . 10 ⊢ (𝑥 = (1^st ‘𝑤) → ⟨1, 𝑥⟩ = ⟨1, (1^st ‘𝑤)⟩)
58	57	preq1d 4700	. . . . . . . . 9 ⊢ (𝑥 = (1^st ‘𝑤) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, (1^st ‘𝑤)⟩, ⟨2, 𝑦⟩})
59		opeq2 4834	. . . . . . . . . 10 ⊢ (𝑦 = (2^nd ‘𝑤) → ⟨2, 𝑦⟩ = ⟨2, (2^nd ‘𝑤)⟩)
60	59	preq2d 4701	. . . . . . . . 9 ⊢ (𝑦 = (2^nd ‘𝑤) → {⟨1, (1^st ‘𝑤)⟩, ⟨2, 𝑦⟩} = {⟨1, (1^st ‘𝑤)⟩, ⟨2, (2^nd ‘𝑤)⟩})
61		prex 5396	. . . . . . . . 9 ⊢ {⟨1, (1^st ‘𝑤)⟩, ⟨2, (2^nd ‘𝑤)⟩} ∈ V
62	58, 60, 1, 61	ovmpo 7580	. . . . . . . 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 596	. . . . . . 7 ⊢ (𝑤 ∈ (ℝ × ℝ) → ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)) = {⟨1, (1^st ‘𝑤)⟩, ⟨2, (2^nd ‘𝑤)⟩})
64	54, 63	eqtrd 2796	. . . . . 6 ⊢ (𝑤 ∈ (ℝ × ℝ) → (𝐹‘𝑤) = {⟨1, (1^st ‘𝑤)⟩, ⟨2, (2^nd ‘𝑤)⟩})
65	17, 64	eqeqan12d 2775	. . . . 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 2775	. . . . . 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 5445	. . . . . 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 290	. . . . 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 297	. . . 4 ⊢ ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
70	69	rgen2 3203	. . 3 ⊢ ∀𝑧 ∈ (ℝ × ℝ)∀𝑤 ∈ (ℝ × ℝ)((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)
71		dff13 7258	. . 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 724	. 2 ⊢ 𝐹:(ℝ × ℝ)–1-1→𝑅
73	19	eleq2i 2853	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑅 ↔ 𝑤 ∈ (ℝ ↑_m {1, 2}))
74		reex 11291	. . . . . . . . 9 ⊢ ℝ ∈ V
75		prex 5396	. . . . . . . . 9 ⊢ {1, 2} ∈ V
76	74, 75	elmap 8899	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ ↑_m {1, 2}) ↔ 𝑤:{1, 2}⟶ℝ)
77		1re 11308	. . . . . . . . 9 ⊢ 1 ∈ ℝ
78		2re 12417	. . . . . . . . 9 ⊢ 2 ∈ ℝ
79		fpr2g 7217	. . . . . . . . 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 705	. . . . . . . 8 ⊢ (𝑤:{1, 2}⟶ℝ ↔ ((𝑤‘1) ∈ ℝ ∧ (𝑤‘2) ∈ ℝ ∧ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩}))
81	73, 76, 80	3bitri 300	. . . . . . 7 ⊢ (𝑤 ∈ 𝑅 ↔ ((𝑤‘1) ∈ ℝ ∧ (𝑤‘2) ∈ ℝ ∧ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩}))
82		opeq2 4834	. . . . . . . . . 10 ⊢ (𝑢 = (𝑤‘1) → ⟨1, 𝑢⟩ = ⟨1, (𝑤‘1)⟩)
83	82	preq1d 4700	. . . . . . . . 9 ⊢ (𝑢 = (𝑤‘1) → {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩})
84	83	eqeq2d 2772	. . . . . . . 8 ⊢ (𝑢 = (𝑤‘1) → (𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} ↔ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩}))
85		opeq2 4834	. . . . . . . . . 10 ⊢ (𝑣 = (𝑤‘2) → ⟨2, 𝑣⟩ = ⟨2, (𝑤‘2)⟩)
86	85	preq2d 4701	. . . . . . . . 9 ⊢ (𝑣 = (𝑤‘2) → {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩} = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩})
87	86	eqeq2d 2772	. . . . . . . 8 ⊢ (𝑣 = (𝑤‘2) → (𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩} ↔ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩}))
88	84, 87	rspc2ev 3589	. . . . . . 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 220	. . . . . 6 ⊢ (𝑤 ∈ 𝑅 → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
90		opeq2 4834	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → ⟨1, 𝑥⟩ = ⟨1, 𝑢⟩)
91	90	preq1d 4700	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑢⟩, ⟨2, 𝑦⟩})
92		opeq2 4834	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → ⟨2, 𝑦⟩ = ⟨2, 𝑣⟩)
93	92	preq2d 4701	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → {⟨1, 𝑢⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
94		prex 5396	. . . . . . . . 9 ⊢ {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} ∈ V
95	91, 93, 1, 94	ovmpo 7580	. . . . . . . 8 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢𝐹𝑣) = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
96	95	eqeq2d 2772	. . . . . . 7 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑤 = (𝑢𝐹𝑣) ↔ 𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩}))
97	96	2rexbiia 3224	. . . . . 6 ⊢ (∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
98	89, 97	sylibr 237	. . . . 5 ⊢ (𝑤 ∈ 𝑅 → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
99		fveq2 6885	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝑧) = (𝐹‘⟨𝑢, 𝑣⟩))
100		df-ov 7423	. . . . . . . 8 ⊢ (𝑢𝐹𝑣) = (𝐹‘⟨𝑢, 𝑣⟩)
101	99, 100	eqtr4di 2814	. . . . . . 7 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝑧) = (𝑢𝐹𝑣))
102	101	eqeq2d 2772	. . . . . 6 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝑤 = (𝐹‘𝑧) ↔ 𝑤 = (𝑢𝐹𝑣)))
103	102	rexxp 5819	. . . . 5 ⊢ (∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
104	98, 103	sylibr 237	. . . 4 ⊢ (𝑤 ∈ 𝑅 → ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧))
105	104	rgen 3079	. . 3 ⊢ ∀𝑤 ∈ 𝑅 ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧)
106		dffo3 7102	. . 3 ⊢ (𝐹:(ℝ × ℝ)–onto→𝑅 ↔ (𝐹:(ℝ × ℝ)⟶𝑅 ∧ ∀𝑤 ∈ 𝑅 ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧)))
107	25, 105, 106	mpbir2an 724	. 2 ⊢ 𝐹:(ℝ × ℝ)–onto→𝑅
108		df-f1o 6545	. 2 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→𝑅 ↔ (𝐹:(ℝ × ℝ)–1-1→𝑅 ∧ 𝐹:(ℝ × ℝ)–onto→𝑅))
109	72, 107, 108	mpbir2an 724	1 ⊢ 𝐹:(ℝ × ℝ)–1-1-onto→𝑅

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∀wral 3077 ∃wrex 3087 {cpr 4586 ⟨cop 4590 × cxp 5649 Fn wfn 6533 ⟶wf 6534 –1-1→wf1 6535 –onto→wfo 6536 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7420 ∈ cmpo 7422 1^st c1st 7999 2^nd c2nd 8000 ↑_m cmap 8847 ℝcr 11199 1c1 11201 2c2 12397

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-po 5559 df-so 5560 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 8001 df-2nd 8002 df-er 8717 df-map 8849 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-2 12405

This theorem is used by: rrx2xpreen 49830 rrx2plordisom 49834

W3C validator