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Theorem rrx2xpref1o 46776
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 8001 . . . 4 𝐹 Fn (ℝ × ℝ)
4 1st2nd2 7959 . . . . . . . . 9 (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
54fveq2d 6846 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩))
6 df-ov 7359 . . . . . . . 8 ((1st𝑧)𝐹(2nd𝑧)) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩)
75, 6eqtr4di 2794 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = ((1st𝑧)𝐹(2nd𝑧)))
8 xp1st 7952 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℝ)
9 xp2nd 7953 . . . . . . . 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 7514 . . . . . . . 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 46771 . . . . . . 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 7065 . . . 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 11150 . . . . . . . . . . . 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 12229 . . . . . . . . . . . 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 12360 . . . . . . . . . . 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 7959 . . . . . . . . 9 (𝑤 ∈ (ℝ × ℝ) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
5251fveq2d 6846 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → (𝐹𝑤) = (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩))
53 df-ov 7359 . . . . . . . 8 ((1st𝑤)𝐹(2nd𝑤)) = (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩)
5452, 53eqtr4di 2794 . . . . . . 7 (𝑤 ∈ (ℝ × ℝ) → (𝐹𝑤) = ((1st𝑤)𝐹(2nd𝑤)))
55 xp1st 7952 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → (1st𝑤) ∈ ℝ)
56 xp2nd 7953 . . . . . . . 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 7514 . . . . . . . 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 7201 . . 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 11141 . . . . . . . . 9 ℝ ∈ V
75 prex 5389 . . . . . . . . 9 {1, 2} ∈ V
7674, 75elmap 8808 . . . . . . . 8 (𝑤 ∈ (ℝ ↑m {1, 2}) ↔ 𝑤:{1, 2}⟶ℝ)
77 1re 11154 . . . . . . . . 9 1 ∈ ℝ
78 2re 12226 . . . . . . . . 9 2 ∈ ℝ
79 fpr2g 7160 . . . . . . . . 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 7514 . . . . . . . 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 7359 . . . . . . . 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 7051 . . 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 7356  cmpo 7358  1st c1st 7918  2nd c2nd 7919  m cmap 8764  cr 11049  1c1 11051  2c2 12207
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 7671  ax-cnex 11106  ax-resscn 11107  ax-1cn 11108  ax-icn 11109  ax-addcl 11110  ax-addrcl 11111  ax-mulcl 11112  ax-mulrcl 11113  ax-mulcom 11114  ax-addass 11115  ax-mulass 11116  ax-distr 11117  ax-i2m1 11118  ax-1ne0 11119  ax-1rid 11120  ax-rnegex 11121  ax-rrecex 11122  ax-cnre 11123  ax-pre-lttri 11124  ax-pre-lttrn 11125  ax-pre-ltadd 11126  ax-pre-mulgt0 11127
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 7312  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7920  df-2nd 7921  df-er 8647  df-map 8766  df-en 8883  df-dom 8884  df-sdom 8885  df-pnf 11190  df-mnf 11191  df-xr 11192  df-ltxr 11193  df-le 11194  df-sub 11386  df-neg 11387  df-2 12215
This theorem is referenced by:  rrx2xpreen  46777  rrx2plordisom  46781
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