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Theorem rrx2xpref1o 45129
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-onto𝑅
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.
StepHypRef Expression
1 rrx2xpref1o.1 . . . . 5 𝐹 = (𝑥 ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, 𝑥⟩, ⟨2, 𝑦⟩})
2 prex 5301 . . . . 5 {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} ∈ V
31, 2fnmpoi 7754 . . . 4 𝐹 Fn (ℝ × ℝ)
4 1st2nd2 7714 . . . . . . . . 9 (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
54fveq2d 6653 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩))
6 df-ov 7142 . . . . . . . 8 ((1st𝑧)𝐹(2nd𝑧)) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩)
75, 6eqtr4di 2854 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = ((1st𝑧)𝐹(2nd𝑧)))
8 xp1st 7707 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℝ)
9 xp2nd 7708 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (2nd𝑧) ∈ ℝ)
10 opeq2 4768 . . . . . . . . . 10 (𝑥 = (1st𝑧) → ⟨1, 𝑥⟩ = ⟨1, (1st𝑧)⟩)
1110preq1d 4638 . . . . . . . . 9 (𝑥 = (1st𝑧) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, (1st𝑧)⟩, ⟨2, 𝑦⟩})
12 opeq2 4768 . . . . . . . . . 10 (𝑦 = (2nd𝑧) → ⟨2, 𝑦⟩ = ⟨2, (2nd𝑧)⟩)
1312preq2d 4639 . . . . . . . . 9 (𝑦 = (2nd𝑧) → {⟨1, (1st𝑧)⟩, ⟨2, 𝑦⟩} = {⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩})
14 prex 5301 . . . . . . . . 9 {⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩} ∈ V
1511, 13, 1, 14ovmpo 7293 . . . . . . . 8 (((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ) → ((1st𝑧)𝐹(2nd𝑧)) = {⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩})
168, 9, 15syl2anc 587 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → ((1st𝑧)𝐹(2nd𝑧)) = {⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩})
177, 16eqtrd 2836 . . . . . 6 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = {⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩})
18 eqid 2801 . . . . . . . 8 {1, 2} = {1, 2}
19 rrx2xpreen.r . . . . . . . 8 𝑅 = (ℝ ↑m {1, 2})
2018, 19prelrrx2 45124 . . . . . . 7 (((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ) → {⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩} ∈ 𝑅)
218, 9, 20syl2anc 587 . . . . . 6 (𝑧 ∈ (ℝ × ℝ) → {⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩} ∈ 𝑅)
2217, 21eqeltrd 2893 . . . . 5 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) ∈ 𝑅)
2322rgen 3119 . . . 4 𝑧 ∈ (ℝ × ℝ)(𝐹𝑧) ∈ 𝑅
24 ffnfv 6863 . . . 4 (𝐹:(ℝ × ℝ)⟶𝑅 ↔ (𝐹 Fn (ℝ × ℝ) ∧ ∀𝑧 ∈ (ℝ × ℝ)(𝐹𝑧) ∈ 𝑅))
253, 23, 24mpbir2an 710 . . 3 𝐹:(ℝ × ℝ)⟶𝑅
26 opex 5324 . . . . . . . 8 ⟨1, (1st𝑧)⟩ ∈ V
27 opex 5324 . . . . . . . 8 ⟨2, (2nd𝑧)⟩ ∈ V
28 opex 5324 . . . . . . . 8 ⟨1, (1st𝑤)⟩ ∈ V
29 opex 5324 . . . . . . . 8 ⟨2, (2nd𝑤)⟩ ∈ V
3026, 27, 28, 29preq12b 4744 . . . . . . 7 ({⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩} = {⟨1, (1st𝑤)⟩, ⟨2, (2nd𝑤)⟩} ↔ ((⟨1, (1st𝑧)⟩ = ⟨1, (1st𝑤)⟩ ∧ ⟨2, (2nd𝑧)⟩ = ⟨2, (2nd𝑤)⟩) ∨ (⟨1, (1st𝑧)⟩ = ⟨2, (2nd𝑤)⟩ ∧ ⟨2, (2nd𝑧)⟩ = ⟨1, (1st𝑤)⟩)))
31 1ex 10630 . . . . . . . . . . . 12 1 ∈ V
32 fvex 6662 . . . . . . . . . . . 12 (1st𝑧) ∈ V
3331, 32opth 5336 . . . . . . . . . . 11 (⟨1, (1st𝑧)⟩ = ⟨1, (1st𝑤)⟩ ↔ (1 = 1 ∧ (1st𝑧) = (1st𝑤)))
3433simprbi 500 . . . . . . . . . 10 (⟨1, (1st𝑧)⟩ = ⟨1, (1st𝑤)⟩ → (1st𝑧) = (1st𝑤))
35 2ex 11706 . . . . . . . . . . . 12 2 ∈ V
36 fvex 6662 . . . . . . . . . . . 12 (2nd𝑧) ∈ V
3735, 36opth 5336 . . . . . . . . . . 11 (⟨2, (2nd𝑧)⟩ = ⟨2, (2nd𝑤)⟩ ↔ (2 = 2 ∧ (2nd𝑧) = (2nd𝑤)))
3837simprbi 500 . . . . . . . . . 10 (⟨2, (2nd𝑧)⟩ = ⟨2, (2nd𝑤)⟩ → (2nd𝑧) = (2nd𝑤))
3934, 38anim12i 615 . . . . . . . . 9 ((⟨1, (1st𝑧)⟩ = ⟨1, (1st𝑤)⟩ ∧ ⟨2, (2nd𝑧)⟩ = ⟨2, (2nd𝑤)⟩) → ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤)))
4039a1d 25 . . . . . . . 8 ((⟨1, (1st𝑧)⟩ = ⟨1, (1st𝑤)⟩ ∧ ⟨2, (2nd𝑧)⟩ = ⟨2, (2nd𝑤)⟩) → ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤))))
4131, 32opth 5336 . . . . . . . . 9 (⟨1, (1st𝑧)⟩ = ⟨2, (2nd𝑤)⟩ ↔ (1 = 2 ∧ (1st𝑧) = (2nd𝑤)))
4235, 36opth 5336 . . . . . . . . 9 (⟨2, (2nd𝑧)⟩ = ⟨1, (1st𝑤)⟩ ↔ (2 = 1 ∧ (2nd𝑧) = (1st𝑤)))
43 1ne2 11837 . . . . . . . . . . 11 1 ≠ 2
44 eqneqall 3001 . . . . . . . . . . 11 (1 = 2 → (1 ≠ 2 → ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤)))))
4543, 44mpi 20 . . . . . . . . . 10 (1 = 2 → ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤))))
4645ad2antrr 725 . . . . . . . . 9 (((1 = 2 ∧ (1st𝑧) = (2nd𝑤)) ∧ (2 = 1 ∧ (2nd𝑧) = (1st𝑤))) → ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤))))
4741, 42, 46syl2anb 600 . . . . . . . 8 ((⟨1, (1st𝑧)⟩ = ⟨2, (2nd𝑤)⟩ ∧ ⟨2, (2nd𝑧)⟩ = ⟨1, (1st𝑤)⟩) → ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤))))
4840, 47jaoi 854 . . . . . . 7 (((⟨1, (1st𝑧)⟩ = ⟨1, (1st𝑤)⟩ ∧ ⟨2, (2nd𝑧)⟩ = ⟨2, (2nd𝑤)⟩) ∨ (⟨1, (1st𝑧)⟩ = ⟨2, (2nd𝑤)⟩ ∧ ⟨2, (2nd𝑧)⟩ = ⟨1, (1st𝑤)⟩)) → ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤))))
4930, 48sylbi 220 . . . . . 6 ({⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩} = {⟨1, (1st𝑤)⟩, ⟨2, (2nd𝑤)⟩} → ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤))))
5049com12 32 . . . . 5 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ({⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩} = {⟨1, (1st𝑤)⟩, ⟨2, (2nd𝑤)⟩} → ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤))))
51 1st2nd2 7714 . . . . . . . . 9 (𝑤 ∈ (ℝ × ℝ) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
5251fveq2d 6653 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → (𝐹𝑤) = (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩))
53 df-ov 7142 . . . . . . . 8 ((1st𝑤)𝐹(2nd𝑤)) = (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩)
5452, 53eqtr4di 2854 . . . . . . 7 (𝑤 ∈ (ℝ × ℝ) → (𝐹𝑤) = ((1st𝑤)𝐹(2nd𝑤)))
55 xp1st 7707 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → (1st𝑤) ∈ ℝ)
56 xp2nd 7708 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → (2nd𝑤) ∈ ℝ)
57 opeq2 4768 . . . . . . . . . 10 (𝑥 = (1st𝑤) → ⟨1, 𝑥⟩ = ⟨1, (1st𝑤)⟩)
5857preq1d 4638 . . . . . . . . 9 (𝑥 = (1st𝑤) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, (1st𝑤)⟩, ⟨2, 𝑦⟩})
59 opeq2 4768 . . . . . . . . . 10 (𝑦 = (2nd𝑤) → ⟨2, 𝑦⟩ = ⟨2, (2nd𝑤)⟩)
6059preq2d 4639 . . . . . . . . 9 (𝑦 = (2nd𝑤) → {⟨1, (1st𝑤)⟩, ⟨2, 𝑦⟩} = {⟨1, (1st𝑤)⟩, ⟨2, (2nd𝑤)⟩})
61 prex 5301 . . . . . . . . 9 {⟨1, (1st𝑤)⟩, ⟨2, (2nd𝑤)⟩} ∈ V
6258, 60, 1, 61ovmpo 7293 . . . . . . . 8 (((1st𝑤) ∈ ℝ ∧ (2nd𝑤) ∈ ℝ) → ((1st𝑤)𝐹(2nd𝑤)) = {⟨1, (1st𝑤)⟩, ⟨2, (2nd𝑤)⟩})
6355, 56, 62syl2anc 587 . . . . . . 7 (𝑤 ∈ (ℝ × ℝ) → ((1st𝑤)𝐹(2nd𝑤)) = {⟨1, (1st𝑤)⟩, ⟨2, (2nd𝑤)⟩})
6454, 63eqtrd 2836 . . . . . 6 (𝑤 ∈ (ℝ × ℝ) → (𝐹𝑤) = {⟨1, (1st𝑤)⟩, ⟨2, (2nd𝑤)⟩})
6517, 64eqeqan12d 2818 . . . . 5 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹𝑧) = (𝐹𝑤) ↔ {⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩} = {⟨1, (1st𝑤)⟩, ⟨2, (2nd𝑤)⟩}))
664, 51eqeqan12d 2818 . . . . . 6 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (𝑧 = 𝑤 ↔ ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩))
6732, 36opth 5336 . . . . . 6 (⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩ ↔ ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤)))
6866, 67syl6bb 290 . . . . 5 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (𝑧 = 𝑤 ↔ ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤))))
6950, 65, 683imtr4d 297 . . . 4 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
7069rgen2 3171 . . 3 𝑧 ∈ (ℝ × ℝ)∀𝑤 ∈ (ℝ × ℝ)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)
71 dff13 6995 . . 3 (𝐹:(ℝ × ℝ)–1-1𝑅 ↔ (𝐹:(ℝ × ℝ)⟶𝑅 ∧ ∀𝑧 ∈ (ℝ × ℝ)∀𝑤 ∈ (ℝ × ℝ)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
7225, 70, 71mpbir2an 710 . 2 𝐹:(ℝ × ℝ)–1-1𝑅
7319eleq2i 2884 . . . . . . . 8 (𝑤𝑅𝑤 ∈ (ℝ ↑m {1, 2}))
74 reex 10621 . . . . . . . . 9 ℝ ∈ V
75 prex 5301 . . . . . . . . 9 {1, 2} ∈ V
7674, 75elmap 8422 . . . . . . . 8 (𝑤 ∈ (ℝ ↑m {1, 2}) ↔ 𝑤:{1, 2}⟶ℝ)
77 1re 10634 . . . . . . . . 9 1 ∈ ℝ
78 2re 11703 . . . . . . . . 9 2 ∈ ℝ
79 fpr2g 6955 . . . . . . . . 9 ((1 ∈ ℝ ∧ 2 ∈ ℝ) → (𝑤:{1, 2}⟶ℝ ↔ ((𝑤‘1) ∈ ℝ ∧ (𝑤‘2) ∈ ℝ ∧ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩})))
8077, 78, 79mp2an 691 . . . . . . . 8 (𝑤:{1, 2}⟶ℝ ↔ ((𝑤‘1) ∈ ℝ ∧ (𝑤‘2) ∈ ℝ ∧ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩}))
8173, 76, 803bitri 300 . . . . . . 7 (𝑤𝑅 ↔ ((𝑤‘1) ∈ ℝ ∧ (𝑤‘2) ∈ ℝ ∧ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩}))
82 opeq2 4768 . . . . . . . . . 10 (𝑢 = (𝑤‘1) → ⟨1, 𝑢⟩ = ⟨1, (𝑤‘1)⟩)
8382preq1d 4638 . . . . . . . . 9 (𝑢 = (𝑤‘1) → {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩})
8483eqeq2d 2812 . . . . . . . 8 (𝑢 = (𝑤‘1) → (𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} ↔ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩}))
85 opeq2 4768 . . . . . . . . . 10 (𝑣 = (𝑤‘2) → ⟨2, 𝑣⟩ = ⟨2, (𝑤‘2)⟩)
8685preq2d 4639 . . . . . . . . 9 (𝑣 = (𝑤‘2) → {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩} = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩})
8786eqeq2d 2812 . . . . . . . 8 (𝑣 = (𝑤‘2) → (𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩} ↔ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩}))
8884, 87rspc2ev 3586 . . . . . . 7 (((𝑤‘1) ∈ ℝ ∧ (𝑤‘2) ∈ ℝ ∧ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩}) → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
8981, 88sylbi 220 . . . . . 6 (𝑤𝑅 → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
90 opeq2 4768 . . . . . . . . . 10 (𝑥 = 𝑢 → ⟨1, 𝑥⟩ = ⟨1, 𝑢⟩)
9190preq1d 4638 . . . . . . . . 9 (𝑥 = 𝑢 → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑢⟩, ⟨2, 𝑦⟩})
92 opeq2 4768 . . . . . . . . . 10 (𝑦 = 𝑣 → ⟨2, 𝑦⟩ = ⟨2, 𝑣⟩)
9392preq2d 4639 . . . . . . . . 9 (𝑦 = 𝑣 → {⟨1, 𝑢⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
94 prex 5301 . . . . . . . . 9 {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} ∈ V
9591, 93, 1, 94ovmpo 7293 . . . . . . . 8 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢𝐹𝑣) = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
9695eqeq2d 2812 . . . . . . 7 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑤 = (𝑢𝐹𝑣) ↔ 𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩}))
97962rexbiia 3260 . . . . . 6 (∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
9889, 97sylibr 237 . . . . 5 (𝑤𝑅 → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
99 fveq2 6649 . . . . . . . 8 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹𝑧) = (𝐹‘⟨𝑢, 𝑣⟩))
100 df-ov 7142 . . . . . . . 8 (𝑢𝐹𝑣) = (𝐹‘⟨𝑢, 𝑣⟩)
10199, 100eqtr4di 2854 . . . . . . 7 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹𝑧) = (𝑢𝐹𝑣))
102101eqeq2d 2812 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝑤 = (𝐹𝑧) ↔ 𝑤 = (𝑢𝐹𝑣)))
103102rexxp 5681 . . . . 5 (∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
10498, 103sylibr 237 . . . 4 (𝑤𝑅 → ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧))
105104rgen 3119 . . 3 𝑤𝑅𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧)
106 dffo3 6849 . . 3 (𝐹:(ℝ × ℝ)–onto𝑅 ↔ (𝐹:(ℝ × ℝ)⟶𝑅 ∧ ∀𝑤𝑅𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧)))
10725, 105, 106mpbir2an 710 . 2 𝐹:(ℝ × ℝ)–onto𝑅
108 df-f1o 6335 . 2 (𝐹:(ℝ × ℝ)–1-1-onto𝑅 ↔ (𝐹:(ℝ × ℝ)–1-1𝑅𝐹:(ℝ × ℝ)–onto𝑅))
10972, 107, 108mpbir2an 710 1 𝐹:(ℝ × ℝ)–1-1-onto𝑅
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wcel 2112  wne 2990  wral 3109  wrex 3110  {cpr 4530  cop 4534   × cxp 5521   Fn wfn 6323  wf 6324  1-1wf1 6325  ontowfo 6326  1-1-ontowf1o 6327  cfv 6328  (class class class)co 7139  cmpo 7141  1st c1st 7673  2nd c2nd 7674  m cmap 8393  cr 10529  1c1 10531  2c2 11684
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-po 5442  df-so 5443  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-2 11692
This theorem is referenced by:  rrx2xpreen  45130  rrx2plordisom  45134
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