rrx2xpref1o - Mathbox for Alexander van der Vekens

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Theorem rrx2xpref1o 48829

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 5373	. . . . 5 ⊢ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} ∈ V
3	1, 2	fnmpoi 8002	. . . 4 ⊢ 𝐹 Fn (ℝ × ℝ)
4		1st2nd2 7960	. . . . . . . . 9 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
5	4	fveq2d 6826	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩))
6		df-ov 7349	. . . . . . . 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 7953	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1^st ‘𝑧) ∈ ℝ)
9		xp2nd 7954	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (2^nd ‘𝑧) ∈ ℝ)
10		opeq2 4823	. . . . . . . . . 10 ⊢ (𝑥 = (1^st ‘𝑧) → ⟨1, 𝑥⟩ = ⟨1, (1^st ‘𝑧)⟩)
11	10	preq1d 4689	. . . . . . . . 9 ⊢ (𝑥 = (1^st ‘𝑧) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, (1^st ‘𝑧)⟩, ⟨2, 𝑦⟩})
12		opeq2 4823	. . . . . . . . . 10 ⊢ (𝑦 = (2^nd ‘𝑧) → ⟨2, 𝑦⟩ = ⟨2, (2^nd ‘𝑧)⟩)
13	12	preq2d 4690	. . . . . . . . 9 ⊢ (𝑦 = (2^nd ‘𝑧) → {⟨1, (1^st ‘𝑧)⟩, ⟨2, 𝑦⟩} = {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩})
14		prex 5373	. . . . . . . . 9 ⊢ {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩} ∈ V
15	11, 13, 1, 14	ovmpo 7506	. . . . . . . 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 2766	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩})
18		eqid 2731	. . . . . . . 8 ⊢ {1, 2} = {1, 2}
19		rrx2xpreen.r	. . . . . . . 8 ⊢ 𝑅 = (ℝ ↑_m {1, 2})
20	18, 19	prelrrx2 48824	. . . . . . 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 2831	. . . . 5 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) ∈ 𝑅)
23	22	rgen 3049	. . . 4 ⊢ ∀𝑧 ∈ (ℝ × ℝ)(𝐹‘𝑧) ∈ 𝑅
24		ffnfv 7052	. . . 4 ⊢ (𝐹:(ℝ × ℝ)⟶𝑅 ↔ (𝐹 Fn (ℝ × ℝ) ∧ ∀𝑧 ∈ (ℝ × ℝ)(𝐹‘𝑧) ∈ 𝑅))
25	3, 23, 24	mpbir2an 711	. . 3 ⊢ 𝐹:(ℝ × ℝ)⟶𝑅
26		opex 5402	. . . . . . . 8 ⊢ ⟨1, (1^st ‘𝑧)⟩ ∈ V
27		opex 5402	. . . . . . . 8 ⊢ ⟨2, (2^nd ‘𝑧)⟩ ∈ V
28		opex 5402	. . . . . . . 8 ⊢ ⟨1, (1^st ‘𝑤)⟩ ∈ V
29		opex 5402	. . . . . . . 8 ⊢ ⟨2, (2^nd ‘𝑤)⟩ ∈ V
30	26, 27, 28, 29	preq12b 4799	. . . . . . 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 11108	. . . . . . . . . . . 12 ⊢ 1 ∈ V
32		fvex 6835	. . . . . . . . . . . 12 ⊢ (1^st ‘𝑧) ∈ V
33	31, 32	opth 5414	. . . . . . . . . . 11 ⊢ (⟨1, (1^st ‘𝑧)⟩ = ⟨1, (1^st ‘𝑤)⟩ ↔ (1 = 1 ∧ (1^st ‘𝑧) = (1^st ‘𝑤)))
34	33	simprbi 496	. . . . . . . . . 10 ⊢ (⟨1, (1^st ‘𝑧)⟩ = ⟨1, (1^st ‘𝑤)⟩ → (1^st ‘𝑧) = (1^st ‘𝑤))
35		2ex 12202	. . . . . . . . . . . 12 ⊢ 2 ∈ V
36		fvex 6835	. . . . . . . . . . . 12 ⊢ (2^nd ‘𝑧) ∈ V
37	35, 36	opth 5414	. . . . . . . . . . 11 ⊢ (⟨2, (2^nd ‘𝑧)⟩ = ⟨2, (2^nd ‘𝑤)⟩ ↔ (2 = 2 ∧ (2^nd ‘𝑧) = (2^nd ‘𝑤)))
38	37	simprbi 496	. . . . . . . . . 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 5414	. . . . . . . . 9 ⊢ (⟨1, (1^st ‘𝑧)⟩ = ⟨2, (2^nd ‘𝑤)⟩ ↔ (1 = 2 ∧ (1^st ‘𝑧) = (2^nd ‘𝑤)))
42	35, 36	opth 5414	. . . . . . . . 9 ⊢ (⟨2, (2^nd ‘𝑧)⟩ = ⟨1, (1^st ‘𝑤)⟩ ↔ (2 = 1 ∧ (2^nd ‘𝑧) = (1^st ‘𝑤)))
43		1ne2 12328	. . . . . . . . . . 11 ⊢ 1 ≠ 2
44		eqneqall 2939	. . . . . . . . . . 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 726	. . . . . . . . 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 857	. . . . . . 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 217	. . . . . 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 7960	. . . . . . . . 9 ⊢ (𝑤 ∈ (ℝ × ℝ) → 𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
52	51	fveq2d 6826	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → (𝐹‘𝑤) = (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩))
53		df-ov 7349	. . . . . . . 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 7953	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → (1^st ‘𝑤) ∈ ℝ)
56		xp2nd 7954	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → (2^nd ‘𝑤) ∈ ℝ)
57		opeq2 4823	. . . . . . . . . 10 ⊢ (𝑥 = (1^st ‘𝑤) → ⟨1, 𝑥⟩ = ⟨1, (1^st ‘𝑤)⟩)
58	57	preq1d 4689	. . . . . . . . 9 ⊢ (𝑥 = (1^st ‘𝑤) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, (1^st ‘𝑤)⟩, ⟨2, 𝑦⟩})
59		opeq2 4823	. . . . . . . . . 10 ⊢ (𝑦 = (2^nd ‘𝑤) → ⟨2, 𝑦⟩ = ⟨2, (2^nd ‘𝑤)⟩)
60	59	preq2d 4690	. . . . . . . . 9 ⊢ (𝑦 = (2^nd ‘𝑤) → {⟨1, (1^st ‘𝑤)⟩, ⟨2, 𝑦⟩} = {⟨1, (1^st ‘𝑤)⟩, ⟨2, (2^nd ‘𝑤)⟩})
61		prex 5373	. . . . . . . . 9 ⊢ {⟨1, (1^st ‘𝑤)⟩, ⟨2, (2^nd ‘𝑤)⟩} ∈ V
62	58, 60, 1, 61	ovmpo 7506	. . . . . . . 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 2766	. . . . . 6 ⊢ (𝑤 ∈ (ℝ × ℝ) → (𝐹‘𝑤) = {⟨1, (1^st ‘𝑤)⟩, ⟨2, (2^nd ‘𝑤)⟩})
65	17, 64	eqeqan12d 2745	. . . . 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 2745	. . . . . 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 5414	. . . . . 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 3172	. . 3 ⊢ ∀𝑧 ∈ (ℝ × ℝ)∀𝑤 ∈ (ℝ × ℝ)((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)
71		dff13 7188	. . 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 711	. 2 ⊢ 𝐹:(ℝ × ℝ)–1-1→𝑅
73	19	eleq2i 2823	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑅 ↔ 𝑤 ∈ (ℝ ↑_m {1, 2}))
74		reex 11097	. . . . . . . . 9 ⊢ ℝ ∈ V
75		prex 5373	. . . . . . . . 9 ⊢ {1, 2} ∈ V
76	74, 75	elmap 8795	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ ↑_m {1, 2}) ↔ 𝑤:{1, 2}⟶ℝ)
77		1re 11112	. . . . . . . . 9 ⊢ 1 ∈ ℝ
78		2re 12199	. . . . . . . . 9 ⊢ 2 ∈ ℝ
79		fpr2g 7145	. . . . . . . . 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 692	. . . . . . . 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 4823	. . . . . . . . . 10 ⊢ (𝑢 = (𝑤‘1) → ⟨1, 𝑢⟩ = ⟨1, (𝑤‘1)⟩)
83	82	preq1d 4689	. . . . . . . . 9 ⊢ (𝑢 = (𝑤‘1) → {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩})
84	83	eqeq2d 2742	. . . . . . . 8 ⊢ (𝑢 = (𝑤‘1) → (𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} ↔ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩}))
85		opeq2 4823	. . . . . . . . . 10 ⊢ (𝑣 = (𝑤‘2) → ⟨2, 𝑣⟩ = ⟨2, (𝑤‘2)⟩)
86	85	preq2d 4690	. . . . . . . . 9 ⊢ (𝑣 = (𝑤‘2) → {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩} = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩})
87	86	eqeq2d 2742	. . . . . . . 8 ⊢ (𝑣 = (𝑤‘2) → (𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩} ↔ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩}))
88	84, 87	rspc2ev 3585	. . . . . . 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 217	. . . . . 6 ⊢ (𝑤 ∈ 𝑅 → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
90		opeq2 4823	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → ⟨1, 𝑥⟩ = ⟨1, 𝑢⟩)
91	90	preq1d 4689	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑢⟩, ⟨2, 𝑦⟩})
92		opeq2 4823	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → ⟨2, 𝑦⟩ = ⟨2, 𝑣⟩)
93	92	preq2d 4690	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → {⟨1, 𝑢⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
94		prex 5373	. . . . . . . . 9 ⊢ {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} ∈ V
95	91, 93, 1, 94	ovmpo 7506	. . . . . . . 8 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢𝐹𝑣) = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
96	95	eqeq2d 2742	. . . . . . 7 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑤 = (𝑢𝐹𝑣) ↔ 𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩}))
97	96	2rexbiia 3193	. . . . . 6 ⊢ (∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
98	89, 97	sylibr 234	. . . . 5 ⊢ (𝑤 ∈ 𝑅 → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
99		fveq2 6822	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝑧) = (𝐹‘⟨𝑢, 𝑣⟩))
100		df-ov 7349	. . . . . . . 8 ⊢ (𝑢𝐹𝑣) = (𝐹‘⟨𝑢, 𝑣⟩)
101	99, 100	eqtr4di 2784	. . . . . . 7 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝑧) = (𝑢𝐹𝑣))
102	101	eqeq2d 2742	. . . . . 6 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝑤 = (𝐹‘𝑧) ↔ 𝑤 = (𝑢𝐹𝑣)))
103	102	rexxp 5781	. . . . 5 ⊢ (∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
104	98, 103	sylibr 234	. . . 4 ⊢ (𝑤 ∈ 𝑅 → ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧))
105	104	rgen 3049	. . 3 ⊢ ∀𝑤 ∈ 𝑅 ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧)
106		dffo3 7035	. . 3 ⊢ (𝐹:(ℝ × ℝ)–onto→𝑅 ↔ (𝐹:(ℝ × ℝ)⟶𝑅 ∧ ∀𝑤 ∈ 𝑅 ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧)))
107	25, 105, 106	mpbir2an 711	. 2 ⊢ 𝐹:(ℝ × ℝ)–onto→𝑅
108		df-f1o 6488	. 2 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→𝑅 ↔ (𝐹:(ℝ × ℝ)–1-1→𝑅 ∧ 𝐹:(ℝ × ℝ)–onto→𝑅))
109	72, 107, 108	mpbir2an 711	1 ⊢ 𝐹:(ℝ × ℝ)–1-1-onto→𝑅

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∃wrex 3056 {cpr 4575 ⟨cop 4579 × cxp 5612 Fn wfn 6476 ⟶wf 6477 –1-1→wf1 6478 –onto→wfo 6479 –1-1-onto→wf1o 6480 ‘cfv 6481 (class class class)co 7346 ∈ cmpo 7348 1^st c1st 7919 2^nd c2nd 7920 ↑_m cmap 8750 ℝcr 11005 1c1 11007 2c2 12180

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-po 5522 df-so 5523 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-1st 7921 df-2nd 7922 df-er 8622 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-2 12188

This theorem is referenced by: rrx2xpreen 48830 rrx2plordisom 48834

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