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Theorem rrx2xpref1o 46794
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 5389 . . . . 5 {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} ∈ V
31, 2fnmpoi 8002 . . . 4 𝐹 Fn (ℝ × ℝ)
4 1st2nd2 7960 . . . . . . . . 9 (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
54fveq2d 6846 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩))
6 df-ov 7360 . . . . . . . 8 ((1st𝑧)𝐹(2nd𝑧)) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩)
75, 6eqtr4di 2794 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = ((1st𝑧)𝐹(2nd𝑧)))
8 xp1st 7953 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℝ)
9 xp2nd 7954 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (2nd𝑧) ∈ ℝ)
10 opeq2 4831 . . . . . . . . . 10 (𝑥 = (1st𝑧) → ⟨1, 𝑥⟩ = ⟨1, (1st𝑧)⟩)
1110preq1d 4700 . . . . . . . . 9 (𝑥 = (1st𝑧) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, (1st𝑧)⟩, ⟨2, 𝑦⟩})
12 opeq2 4831 . . . . . . . . . 10 (𝑦 = (2nd𝑧) → ⟨2, 𝑦⟩ = ⟨2, (2nd𝑧)⟩)
1312preq2d 4701 . . . . . . . . 9 (𝑦 = (2nd𝑧) → {⟨1, (1st𝑧)⟩, ⟨2, 𝑦⟩} = {⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩})
14 prex 5389 . . . . . . . . 9 {⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩} ∈ V
1511, 13, 1, 14ovmpo 7515 . . . . . . . 8 (((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ) → ((1st𝑧)𝐹(2nd𝑧)) = {⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩})
168, 9, 15syl2anc 584 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → ((1st𝑧)𝐹(2nd𝑧)) = {⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩})
177, 16eqtrd 2776 . . . . . 6 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = {⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩})
18 eqid 2736 . . . . . . . 8 {1, 2} = {1, 2}
19 rrx2xpreen.r . . . . . . . 8 𝑅 = (ℝ ↑m {1, 2})
2018, 19prelrrx2 46789 . . . . . . 7 (((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ) → {⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩} ∈ 𝑅)
218, 9, 20syl2anc 584 . . . . . 6 (𝑧 ∈ (ℝ × ℝ) → {⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩} ∈ 𝑅)
2217, 21eqeltrd 2838 . . . . 5 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) ∈ 𝑅)
2322rgen 3066 . . . 4 𝑧 ∈ (ℝ × ℝ)(𝐹𝑧) ∈ 𝑅
24 ffnfv 7066 . . . 4 (𝐹:(ℝ × ℝ)⟶𝑅 ↔ (𝐹 Fn (ℝ × ℝ) ∧ ∀𝑧 ∈ (ℝ × ℝ)(𝐹𝑧) ∈ 𝑅))
253, 23, 24mpbir2an 709 . . 3 𝐹:(ℝ × ℝ)⟶𝑅
26 opex 5421 . . . . . . . 8 ⟨1, (1st𝑧)⟩ ∈ V
27 opex 5421 . . . . . . . 8 ⟨2, (2nd𝑧)⟩ ∈ V
28 opex 5421 . . . . . . . 8 ⟨1, (1st𝑤)⟩ ∈ V
29 opex 5421 . . . . . . . 8 ⟨2, (2nd𝑤)⟩ ∈ V
3026, 27, 28, 29preq12b 4808 . . . . . . 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 11151 . . . . . . . . . . . 12 1 ∈ V
32 fvex 6855 . . . . . . . . . . . 12 (1st𝑧) ∈ V
3331, 32opth 5433 . . . . . . . . . . 11 (⟨1, (1st𝑧)⟩ = ⟨1, (1st𝑤)⟩ ↔ (1 = 1 ∧ (1st𝑧) = (1st𝑤)))
3433simprbi 497 . . . . . . . . . 10 (⟨1, (1st𝑧)⟩ = ⟨1, (1st𝑤)⟩ → (1st𝑧) = (1st𝑤))
35 2ex 12230 . . . . . . . . . . . 12 2 ∈ V
36 fvex 6855 . . . . . . . . . . . 12 (2nd𝑧) ∈ V
3735, 36opth 5433 . . . . . . . . . . 11 (⟨2, (2nd𝑧)⟩ = ⟨2, (2nd𝑤)⟩ ↔ (2 = 2 ∧ (2nd𝑧) = (2nd𝑤)))
3837simprbi 497 . . . . . . . . . 10 (⟨2, (2nd𝑧)⟩ = ⟨2, (2nd𝑤)⟩ → (2nd𝑧) = (2nd𝑤))
3934, 38anim12i 613 . . . . . . . . 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 5433 . . . . . . . . 9 (⟨1, (1st𝑧)⟩ = ⟨2, (2nd𝑤)⟩ ↔ (1 = 2 ∧ (1st𝑧) = (2nd𝑤)))
4235, 36opth 5433 . . . . . . . . 9 (⟨2, (2nd𝑧)⟩ = ⟨1, (1st𝑤)⟩ ↔ (2 = 1 ∧ (2nd𝑧) = (1st𝑤)))
43 1ne2 12361 . . . . . . . . . . 11 1 ≠ 2
44 eqneqall 2954 . . . . . . . . . . 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 724 . . . . . . . . 9 (((1 = 2 ∧ (1st𝑧) = (2nd𝑤)) ∧ (2 = 1 ∧ (2nd𝑧) = (1st𝑤))) → ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤))))
4741, 42, 46syl2anb 598 . . . . . . . 8 ((⟨1, (1st𝑧)⟩ = ⟨2, (2nd𝑤)⟩ ∧ ⟨2, (2nd𝑧)⟩ = ⟨1, (1st𝑤)⟩) → ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤))))
4840, 47jaoi 855 . . . . . . 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 216 . . . . . 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 7960 . . . . . . . . 9 (𝑤 ∈ (ℝ × ℝ) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
5251fveq2d 6846 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → (𝐹𝑤) = (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩))
53 df-ov 7360 . . . . . . . 8 ((1st𝑤)𝐹(2nd𝑤)) = (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩)
5452, 53eqtr4di 2794 . . . . . . 7 (𝑤 ∈ (ℝ × ℝ) → (𝐹𝑤) = ((1st𝑤)𝐹(2nd𝑤)))
55 xp1st 7953 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → (1st𝑤) ∈ ℝ)
56 xp2nd 7954 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → (2nd𝑤) ∈ ℝ)
57 opeq2 4831 . . . . . . . . . 10 (𝑥 = (1st𝑤) → ⟨1, 𝑥⟩ = ⟨1, (1st𝑤)⟩)
5857preq1d 4700 . . . . . . . . 9 (𝑥 = (1st𝑤) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, (1st𝑤)⟩, ⟨2, 𝑦⟩})
59 opeq2 4831 . . . . . . . . . 10 (𝑦 = (2nd𝑤) → ⟨2, 𝑦⟩ = ⟨2, (2nd𝑤)⟩)
6059preq2d 4701 . . . . . . . . 9 (𝑦 = (2nd𝑤) → {⟨1, (1st𝑤)⟩, ⟨2, 𝑦⟩} = {⟨1, (1st𝑤)⟩, ⟨2, (2nd𝑤)⟩})
61 prex 5389 . . . . . . . . 9 {⟨1, (1st𝑤)⟩, ⟨2, (2nd𝑤)⟩} ∈ V
6258, 60, 1, 61ovmpo 7515 . . . . . . . 8 (((1st𝑤) ∈ ℝ ∧ (2nd𝑤) ∈ ℝ) → ((1st𝑤)𝐹(2nd𝑤)) = {⟨1, (1st𝑤)⟩, ⟨2, (2nd𝑤)⟩})
6355, 56, 62syl2anc 584 . . . . . . 7 (𝑤 ∈ (ℝ × ℝ) → ((1st𝑤)𝐹(2nd𝑤)) = {⟨1, (1st𝑤)⟩, ⟨2, (2nd𝑤)⟩})
6454, 63eqtrd 2776 . . . . . 6 (𝑤 ∈ (ℝ × ℝ) → (𝐹𝑤) = {⟨1, (1st𝑤)⟩, ⟨2, (2nd𝑤)⟩})
6517, 64eqeqan12d 2750 . . . . 5 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹𝑧) = (𝐹𝑤) ↔ {⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩} = {⟨1, (1st𝑤)⟩, ⟨2, (2nd𝑤)⟩}))
664, 51eqeqan12d 2750 . . . . . 6 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (𝑧 = 𝑤 ↔ ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩))
6732, 36opth 5433 . . . . . 6 (⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩ ↔ ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤)))
6866, 67bitrdi 286 . . . . 5 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (𝑧 = 𝑤 ↔ ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤))))
6950, 65, 683imtr4d 293 . . . 4 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
7069rgen2 3194 . . 3 𝑧 ∈ (ℝ × ℝ)∀𝑤 ∈ (ℝ × ℝ)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)
71 dff13 7202 . . 3 (𝐹:(ℝ × ℝ)–1-1𝑅 ↔ (𝐹:(ℝ × ℝ)⟶𝑅 ∧ ∀𝑧 ∈ (ℝ × ℝ)∀𝑤 ∈ (ℝ × ℝ)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
7225, 70, 71mpbir2an 709 . 2 𝐹:(ℝ × ℝ)–1-1𝑅
7319eleq2i 2829 . . . . . . . 8 (𝑤𝑅𝑤 ∈ (ℝ ↑m {1, 2}))
74 reex 11142 . . . . . . . . 9 ℝ ∈ V
75 prex 5389 . . . . . . . . 9 {1, 2} ∈ V
7674, 75elmap 8809 . . . . . . . 8 (𝑤 ∈ (ℝ ↑m {1, 2}) ↔ 𝑤:{1, 2}⟶ℝ)
77 1re 11155 . . . . . . . . 9 1 ∈ ℝ
78 2re 12227 . . . . . . . . 9 2 ∈ ℝ
79 fpr2g 7161 . . . . . . . . 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 690 . . . . . . . 8 (𝑤:{1, 2}⟶ℝ ↔ ((𝑤‘1) ∈ ℝ ∧ (𝑤‘2) ∈ ℝ ∧ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩}))
8173, 76, 803bitri 296 . . . . . . 7 (𝑤𝑅 ↔ ((𝑤‘1) ∈ ℝ ∧ (𝑤‘2) ∈ ℝ ∧ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩}))
82 opeq2 4831 . . . . . . . . . 10 (𝑢 = (𝑤‘1) → ⟨1, 𝑢⟩ = ⟨1, (𝑤‘1)⟩)
8382preq1d 4700 . . . . . . . . 9 (𝑢 = (𝑤‘1) → {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩})
8483eqeq2d 2747 . . . . . . . 8 (𝑢 = (𝑤‘1) → (𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} ↔ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩}))
85 opeq2 4831 . . . . . . . . . 10 (𝑣 = (𝑤‘2) → ⟨2, 𝑣⟩ = ⟨2, (𝑤‘2)⟩)
8685preq2d 4701 . . . . . . . . 9 (𝑣 = (𝑤‘2) → {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩} = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩})
8786eqeq2d 2747 . . . . . . . 8 (𝑣 = (𝑤‘2) → (𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩} ↔ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩}))
8884, 87rspc2ev 3592 . . . . . . 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 216 . . . . . 6 (𝑤𝑅 → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
90 opeq2 4831 . . . . . . . . . 10 (𝑥 = 𝑢 → ⟨1, 𝑥⟩ = ⟨1, 𝑢⟩)
9190preq1d 4700 . . . . . . . . 9 (𝑥 = 𝑢 → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑢⟩, ⟨2, 𝑦⟩})
92 opeq2 4831 . . . . . . . . . 10 (𝑦 = 𝑣 → ⟨2, 𝑦⟩ = ⟨2, 𝑣⟩)
9392preq2d 4701 . . . . . . . . 9 (𝑦 = 𝑣 → {⟨1, 𝑢⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
94 prex 5389 . . . . . . . . 9 {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} ∈ V
9591, 93, 1, 94ovmpo 7515 . . . . . . . 8 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢𝐹𝑣) = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
9695eqeq2d 2747 . . . . . . 7 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑤 = (𝑢𝐹𝑣) ↔ 𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩}))
97962rexbiia 3209 . . . . . 6 (∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
9889, 97sylibr 233 . . . . 5 (𝑤𝑅 → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
99 fveq2 6842 . . . . . . . 8 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹𝑧) = (𝐹‘⟨𝑢, 𝑣⟩))
100 df-ov 7360 . . . . . . . 8 (𝑢𝐹𝑣) = (𝐹‘⟨𝑢, 𝑣⟩)
10199, 100eqtr4di 2794 . . . . . . 7 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹𝑧) = (𝑢𝐹𝑣))
102101eqeq2d 2747 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝑤 = (𝐹𝑧) ↔ 𝑤 = (𝑢𝐹𝑣)))
103102rexxp 5798 . . . . 5 (∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
10498, 103sylibr 233 . . . 4 (𝑤𝑅 → ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧))
105104rgen 3066 . . 3 𝑤𝑅𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧)
106 dffo3 7052 . . 3 (𝐹:(ℝ × ℝ)–onto𝑅 ↔ (𝐹:(ℝ × ℝ)⟶𝑅 ∧ ∀𝑤𝑅𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧)))
10725, 105, 106mpbir2an 709 . 2 𝐹:(ℝ × ℝ)–onto𝑅
108 df-f1o 6503 . 2 (𝐹:(ℝ × ℝ)–1-1-onto𝑅 ↔ (𝐹:(ℝ × ℝ)–1-1𝑅𝐹:(ℝ × ℝ)–onto𝑅))
10972, 107, 108mpbir2an 709 1 𝐹:(ℝ × ℝ)–1-1-onto𝑅
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  {cpr 4588  cop 4592   × cxp 5631   Fn wfn 6491  wf 6492  1-1wf1 6493  ontowfo 6494  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  cmpo 7359  1st c1st 7919  2nd c2nd 7920  m cmap 8765  cr 11050  1c1 11052  2c2 12208
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-so 5546  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-2 12216
This theorem is referenced by:  rrx2xpreen  46795  rrx2plordisom  46799
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