rrx2xpref1o - Mathbox for Alexander van der Vekens

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Theorem rrx2xpref1o 49194

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 5380	. . . . 5 ⊢ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} ∈ V
3	1, 2	fnmpoi 8023	. . . 4 ⊢ 𝐹 Fn (ℝ × ℝ)
4		1st2nd2 7981	. . . . . . . . 9 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
5	4	fveq2d 6844	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩))
6		df-ov 7370	. . . . . . . 8 ⊢ ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)) = (𝐹‘⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
7	5, 6	eqtr4di 2789	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)))
8		xp1st 7974	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1^st ‘𝑧) ∈ ℝ)
9		xp2nd 7975	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (2^nd ‘𝑧) ∈ ℝ)
10		opeq2 4817	. . . . . . . . . 10 ⊢ (𝑥 = (1^st ‘𝑧) → ⟨1, 𝑥⟩ = ⟨1, (1^st ‘𝑧)⟩)
11	10	preq1d 4683	. . . . . . . . 9 ⊢ (𝑥 = (1^st ‘𝑧) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, (1^st ‘𝑧)⟩, ⟨2, 𝑦⟩})
12		opeq2 4817	. . . . . . . . . 10 ⊢ (𝑦 = (2^nd ‘𝑧) → ⟨2, 𝑦⟩ = ⟨2, (2^nd ‘𝑧)⟩)
13	12	preq2d 4684	. . . . . . . . 9 ⊢ (𝑦 = (2^nd ‘𝑧) → {⟨1, (1^st ‘𝑧)⟩, ⟨2, 𝑦⟩} = {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩})
14		prex 5380	. . . . . . . . 9 ⊢ {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩} ∈ V
15	11, 13, 1, 14	ovmpo 7527	. . . . . . . 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 585	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → ((1^st ‘𝑧)𝐹(2^nd ‘𝑧)) = {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩})
17	7, 16	eqtrd 2771	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) = {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩})
18		eqid 2736	. . . . . . . 8 ⊢ {1, 2} = {1, 2}
19		rrx2xpreen.r	. . . . . . . 8 ⊢ 𝑅 = (ℝ ↑_m {1, 2})
20	18, 19	prelrrx2 49189	. . . . . . 7 ⊢ (((1^st ‘𝑧) ∈ ℝ ∧ (2^nd ‘𝑧) ∈ ℝ) → {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩} ∈ 𝑅)
21	8, 9, 20	syl2anc 585	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → {⟨1, (1^st ‘𝑧)⟩, ⟨2, (2^nd ‘𝑧)⟩} ∈ 𝑅)
22	17, 21	eqeltrd 2836	. . . . 5 ⊢ (𝑧 ∈ (ℝ × ℝ) → (𝐹‘𝑧) ∈ 𝑅)
23	22	rgen 3053	. . . 4 ⊢ ∀𝑧 ∈ (ℝ × ℝ)(𝐹‘𝑧) ∈ 𝑅
24		ffnfv 7071	. . . 4 ⊢ (𝐹:(ℝ × ℝ)⟶𝑅 ↔ (𝐹 Fn (ℝ × ℝ) ∧ ∀𝑧 ∈ (ℝ × ℝ)(𝐹‘𝑧) ∈ 𝑅))
25	3, 23, 24	mpbir2an 712	. . 3 ⊢ 𝐹:(ℝ × ℝ)⟶𝑅
26		opex 5416	. . . . . . . 8 ⊢ ⟨1, (1^st ‘𝑧)⟩ ∈ V
27		opex 5416	. . . . . . . 8 ⊢ ⟨2, (2^nd ‘𝑧)⟩ ∈ V
28		opex 5416	. . . . . . . 8 ⊢ ⟨1, (1^st ‘𝑤)⟩ ∈ V
29		opex 5416	. . . . . . . 8 ⊢ ⟨2, (2^nd ‘𝑤)⟩ ∈ V
30	26, 27, 28, 29	preq12b 4793	. . . . . . 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 11140	. . . . . . . . . . . 12 ⊢ 1 ∈ V
32		fvex 6853	. . . . . . . . . . . 12 ⊢ (1^st ‘𝑧) ∈ V
33	31, 32	opth 5429	. . . . . . . . . . 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 12258	. . . . . . . . . . . 12 ⊢ 2 ∈ V
36		fvex 6853	. . . . . . . . . . . 12 ⊢ (2^nd ‘𝑧) ∈ V
37	35, 36	opth 5429	. . . . . . . . . . 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 614	. . . . . . . . 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 5429	. . . . . . . . 9 ⊢ (⟨1, (1^st ‘𝑧)⟩ = ⟨2, (2^nd ‘𝑤)⟩ ↔ (1 = 2 ∧ (1^st ‘𝑧) = (2^nd ‘𝑤)))
42	35, 36	opth 5429	. . . . . . . . 9 ⊢ (⟨2, (2^nd ‘𝑧)⟩ = ⟨1, (1^st ‘𝑤)⟩ ↔ (2 = 1 ∧ (2^nd ‘𝑧) = (1^st ‘𝑤)))
43		1ne2 12384	. . . . . . . . . . 11 ⊢ 1 ≠ 2
44		eqneqall 2943	. . . . . . . . . . 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 727	. . . . . . . . 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 599	. . . . . . . 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 858	. . . . . . 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 7981	. . . . . . . . 9 ⊢ (𝑤 ∈ (ℝ × ℝ) → 𝑤 = ⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
52	51	fveq2d 6844	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → (𝐹‘𝑤) = (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩))
53		df-ov 7370	. . . . . . . 8 ⊢ ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)) = (𝐹‘⟨(1^st ‘𝑤), (2^nd ‘𝑤)⟩)
54	52, 53	eqtr4di 2789	. . . . . . 7 ⊢ (𝑤 ∈ (ℝ × ℝ) → (𝐹‘𝑤) = ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)))
55		xp1st 7974	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → (1^st ‘𝑤) ∈ ℝ)
56		xp2nd 7975	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ × ℝ) → (2^nd ‘𝑤) ∈ ℝ)
57		opeq2 4817	. . . . . . . . . 10 ⊢ (𝑥 = (1^st ‘𝑤) → ⟨1, 𝑥⟩ = ⟨1, (1^st ‘𝑤)⟩)
58	57	preq1d 4683	. . . . . . . . 9 ⊢ (𝑥 = (1^st ‘𝑤) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, (1^st ‘𝑤)⟩, ⟨2, 𝑦⟩})
59		opeq2 4817	. . . . . . . . . 10 ⊢ (𝑦 = (2^nd ‘𝑤) → ⟨2, 𝑦⟩ = ⟨2, (2^nd ‘𝑤)⟩)
60	59	preq2d 4684	. . . . . . . . 9 ⊢ (𝑦 = (2^nd ‘𝑤) → {⟨1, (1^st ‘𝑤)⟩, ⟨2, 𝑦⟩} = {⟨1, (1^st ‘𝑤)⟩, ⟨2, (2^nd ‘𝑤)⟩})
61		prex 5380	. . . . . . . . 9 ⊢ {⟨1, (1^st ‘𝑤)⟩, ⟨2, (2^nd ‘𝑤)⟩} ∈ V
62	58, 60, 1, 61	ovmpo 7527	. . . . . . . 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 585	. . . . . . 7 ⊢ (𝑤 ∈ (ℝ × ℝ) → ((1^st ‘𝑤)𝐹(2^nd ‘𝑤)) = {⟨1, (1^st ‘𝑤)⟩, ⟨2, (2^nd ‘𝑤)⟩})
64	54, 63	eqtrd 2771	. . . . . 6 ⊢ (𝑤 ∈ (ℝ × ℝ) → (𝐹‘𝑤) = {⟨1, (1^st ‘𝑤)⟩, ⟨2, (2^nd ‘𝑤)⟩})
65	17, 64	eqeqan12d 2750	. . . . 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 2750	. . . . . 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 5429	. . . . . 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 3177	. . 3 ⊢ ∀𝑧 ∈ (ℝ × ℝ)∀𝑤 ∈ (ℝ × ℝ)((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤)
71		dff13 7209	. . 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 712	. 2 ⊢ 𝐹:(ℝ × ℝ)–1-1→𝑅
73	19	eleq2i 2828	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑅 ↔ 𝑤 ∈ (ℝ ↑_m {1, 2}))
74		reex 11129	. . . . . . . . 9 ⊢ ℝ ∈ V
75		prex 5380	. . . . . . . . 9 ⊢ {1, 2} ∈ V
76	74, 75	elmap 8819	. . . . . . . 8 ⊢ (𝑤 ∈ (ℝ ↑_m {1, 2}) ↔ 𝑤:{1, 2}⟶ℝ)
77		1re 11144	. . . . . . . . 9 ⊢ 1 ∈ ℝ
78		2re 12255	. . . . . . . . 9 ⊢ 2 ∈ ℝ
79		fpr2g 7166	. . . . . . . . 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 693	. . . . . . . 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 4817	. . . . . . . . . 10 ⊢ (𝑢 = (𝑤‘1) → ⟨1, 𝑢⟩ = ⟨1, (𝑤‘1)⟩)
83	82	preq1d 4683	. . . . . . . . 9 ⊢ (𝑢 = (𝑤‘1) → {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩})
84	83	eqeq2d 2747	. . . . . . . 8 ⊢ (𝑢 = (𝑤‘1) → (𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} ↔ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩}))
85		opeq2 4817	. . . . . . . . . 10 ⊢ (𝑣 = (𝑤‘2) → ⟨2, 𝑣⟩ = ⟨2, (𝑤‘2)⟩)
86	85	preq2d 4684	. . . . . . . . 9 ⊢ (𝑣 = (𝑤‘2) → {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩} = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩})
87	86	eqeq2d 2747	. . . . . . . 8 ⊢ (𝑣 = (𝑤‘2) → (𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩} ↔ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩}))
88	84, 87	rspc2ev 3577	. . . . . . 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 4817	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → ⟨1, 𝑥⟩ = ⟨1, 𝑢⟩)
91	90	preq1d 4683	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑢⟩, ⟨2, 𝑦⟩})
92		opeq2 4817	. . . . . . . . . 10 ⊢ (𝑦 = 𝑣 → ⟨2, 𝑦⟩ = ⟨2, 𝑣⟩)
93	92	preq2d 4684	. . . . . . . . 9 ⊢ (𝑦 = 𝑣 → {⟨1, 𝑢⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
94		prex 5380	. . . . . . . . 9 ⊢ {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} ∈ V
95	91, 93, 1, 94	ovmpo 7527	. . . . . . . 8 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢𝐹𝑣) = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
96	95	eqeq2d 2747	. . . . . . 7 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑤 = (𝑢𝐹𝑣) ↔ 𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩}))
97	96	2rexbiia 3198	. . . . . 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 6840	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝑧) = (𝐹‘⟨𝑢, 𝑣⟩))
100		df-ov 7370	. . . . . . . 8 ⊢ (𝑢𝐹𝑣) = (𝐹‘⟨𝑢, 𝑣⟩)
101	99, 100	eqtr4di 2789	. . . . . . 7 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹‘𝑧) = (𝑢𝐹𝑣))
102	101	eqeq2d 2747	. . . . . 6 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ → (𝑤 = (𝐹‘𝑧) ↔ 𝑤 = (𝑢𝐹𝑣)))
103	102	rexxp 5797	. . . . 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 3053	. . 3 ⊢ ∀𝑤 ∈ 𝑅 ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧)
106		dffo3 7054	. . 3 ⊢ (𝐹:(ℝ × ℝ)–onto→𝑅 ↔ (𝐹:(ℝ × ℝ)⟶𝑅 ∧ ∀𝑤 ∈ 𝑅 ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹‘𝑧)))
107	25, 105, 106	mpbir2an 712	. 2 ⊢ 𝐹:(ℝ × ℝ)–onto→𝑅
108		df-f1o 6505	. 2 ⊢ (𝐹:(ℝ × ℝ)–1-1-onto→𝑅 ↔ (𝐹:(ℝ × ℝ)–1-1→𝑅 ∧ 𝐹:(ℝ × ℝ)–onto→𝑅))
109	72, 107, 108	mpbir2an 712	1 ⊢ 𝐹:(ℝ × ℝ)–1-1-onto→𝑅

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∀wral 3051 ∃wrex 3061 {cpr 4569 ⟨cop 4573 × cxp 5629 Fn wfn 6493 ⟶wf 6494 –1-1→wf1 6495 –onto→wfo 6496 –1-1-onto→wf1o 6497 ‘cfv 6498 (class class class)co 7367 ∈ cmpo 7369 1^st c1st 7940 2^nd c2nd 7941 ↑_m cmap 8773 ℝcr 11037 1c1 11039 2c2 12236

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-po 5539 df-so 5540 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-1st 7942 df-2nd 7943 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-2 12244

This theorem is referenced by: rrx2xpreen 49195 rrx2plordisom 49199

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