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Theorem rrx2xpref1o 48568
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 5443 . . . . 5 {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} ∈ V
31, 2fnmpoi 8094 . . . 4 𝐹 Fn (ℝ × ℝ)
4 1st2nd2 8052 . . . . . . . . 9 (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
54fveq2d 6911 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩))
6 df-ov 7434 . . . . . . . 8 ((1st𝑧)𝐹(2nd𝑧)) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩)
75, 6eqtr4di 2793 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = ((1st𝑧)𝐹(2nd𝑧)))
8 xp1st 8045 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℝ)
9 xp2nd 8046 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (2nd𝑧) ∈ ℝ)
10 opeq2 4879 . . . . . . . . . 10 (𝑥 = (1st𝑧) → ⟨1, 𝑥⟩ = ⟨1, (1st𝑧)⟩)
1110preq1d 4744 . . . . . . . . 9 (𝑥 = (1st𝑧) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, (1st𝑧)⟩, ⟨2, 𝑦⟩})
12 opeq2 4879 . . . . . . . . . 10 (𝑦 = (2nd𝑧) → ⟨2, 𝑦⟩ = ⟨2, (2nd𝑧)⟩)
1312preq2d 4745 . . . . . . . . 9 (𝑦 = (2nd𝑧) → {⟨1, (1st𝑧)⟩, ⟨2, 𝑦⟩} = {⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩})
14 prex 5443 . . . . . . . . 9 {⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩} ∈ V
1511, 13, 1, 14ovmpo 7593 . . . . . . . 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 2775 . . . . . 6 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = {⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩})
18 eqid 2735 . . . . . . . 8 {1, 2} = {1, 2}
19 rrx2xpreen.r . . . . . . . 8 𝑅 = (ℝ ↑m {1, 2})
2018, 19prelrrx2 48563 . . . . . . 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 2839 . . . . 5 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) ∈ 𝑅)
2322rgen 3061 . . . 4 𝑧 ∈ (ℝ × ℝ)(𝐹𝑧) ∈ 𝑅
24 ffnfv 7139 . . . 4 (𝐹:(ℝ × ℝ)⟶𝑅 ↔ (𝐹 Fn (ℝ × ℝ) ∧ ∀𝑧 ∈ (ℝ × ℝ)(𝐹𝑧) ∈ 𝑅))
253, 23, 24mpbir2an 711 . . 3 𝐹:(ℝ × ℝ)⟶𝑅
26 opex 5475 . . . . . . . 8 ⟨1, (1st𝑧)⟩ ∈ V
27 opex 5475 . . . . . . . 8 ⟨2, (2nd𝑧)⟩ ∈ V
28 opex 5475 . . . . . . . 8 ⟨1, (1st𝑤)⟩ ∈ V
29 opex 5475 . . . . . . . 8 ⟨2, (2nd𝑤)⟩ ∈ V
3026, 27, 28, 29preq12b 4855 . . . . . . 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 11255 . . . . . . . . . . . 12 1 ∈ V
32 fvex 6920 . . . . . . . . . . . 12 (1st𝑧) ∈ V
3331, 32opth 5487 . . . . . . . . . . 11 (⟨1, (1st𝑧)⟩ = ⟨1, (1st𝑤)⟩ ↔ (1 = 1 ∧ (1st𝑧) = (1st𝑤)))
3433simprbi 496 . . . . . . . . . 10 (⟨1, (1st𝑧)⟩ = ⟨1, (1st𝑤)⟩ → (1st𝑧) = (1st𝑤))
35 2ex 12341 . . . . . . . . . . . 12 2 ∈ V
36 fvex 6920 . . . . . . . . . . . 12 (2nd𝑧) ∈ V
3735, 36opth 5487 . . . . . . . . . . 11 (⟨2, (2nd𝑧)⟩ = ⟨2, (2nd𝑤)⟩ ↔ (2 = 2 ∧ (2nd𝑧) = (2nd𝑤)))
3837simprbi 496 . . . . . . . . . 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 5487 . . . . . . . . 9 (⟨1, (1st𝑧)⟩ = ⟨2, (2nd𝑤)⟩ ↔ (1 = 2 ∧ (1st𝑧) = (2nd𝑤)))
4235, 36opth 5487 . . . . . . . . 9 (⟨2, (2nd𝑧)⟩ = ⟨1, (1st𝑤)⟩ ↔ (2 = 1 ∧ (2nd𝑧) = (1st𝑤)))
43 1ne2 12472 . . . . . . . . . . 11 1 ≠ 2
44 eqneqall 2949 . . . . . . . . . . 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 726 . . . . . . . . 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 857 . . . . . . 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 217 . . . . . 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 8052 . . . . . . . . 9 (𝑤 ∈ (ℝ × ℝ) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
5251fveq2d 6911 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → (𝐹𝑤) = (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩))
53 df-ov 7434 . . . . . . . 8 ((1st𝑤)𝐹(2nd𝑤)) = (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩)
5452, 53eqtr4di 2793 . . . . . . 7 (𝑤 ∈ (ℝ × ℝ) → (𝐹𝑤) = ((1st𝑤)𝐹(2nd𝑤)))
55 xp1st 8045 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → (1st𝑤) ∈ ℝ)
56 xp2nd 8046 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → (2nd𝑤) ∈ ℝ)
57 opeq2 4879 . . . . . . . . . 10 (𝑥 = (1st𝑤) → ⟨1, 𝑥⟩ = ⟨1, (1st𝑤)⟩)
5857preq1d 4744 . . . . . . . . 9 (𝑥 = (1st𝑤) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, (1st𝑤)⟩, ⟨2, 𝑦⟩})
59 opeq2 4879 . . . . . . . . . 10 (𝑦 = (2nd𝑤) → ⟨2, 𝑦⟩ = ⟨2, (2nd𝑤)⟩)
6059preq2d 4745 . . . . . . . . 9 (𝑦 = (2nd𝑤) → {⟨1, (1st𝑤)⟩, ⟨2, 𝑦⟩} = {⟨1, (1st𝑤)⟩, ⟨2, (2nd𝑤)⟩})
61 prex 5443 . . . . . . . . 9 {⟨1, (1st𝑤)⟩, ⟨2, (2nd𝑤)⟩} ∈ V
6258, 60, 1, 61ovmpo 7593 . . . . . . . 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 2775 . . . . . 6 (𝑤 ∈ (ℝ × ℝ) → (𝐹𝑤) = {⟨1, (1st𝑤)⟩, ⟨2, (2nd𝑤)⟩})
6517, 64eqeqan12d 2749 . . . . 5 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹𝑧) = (𝐹𝑤) ↔ {⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩} = {⟨1, (1st𝑤)⟩, ⟨2, (2nd𝑤)⟩}))
664, 51eqeqan12d 2749 . . . . . 6 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (𝑧 = 𝑤 ↔ ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩))
6732, 36opth 5487 . . . . . 6 (⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩ ↔ ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤)))
6866, 67bitrdi 287 . . . . 5 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (𝑧 = 𝑤 ↔ ((1st𝑧) = (1st𝑤) ∧ (2nd𝑧) = (2nd𝑤))))
6950, 65, 683imtr4d 294 . . . 4 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
7069rgen2 3197 . . 3 𝑧 ∈ (ℝ × ℝ)∀𝑤 ∈ (ℝ × ℝ)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)
71 dff13 7275 . . 3 (𝐹:(ℝ × ℝ)–1-1𝑅 ↔ (𝐹:(ℝ × ℝ)⟶𝑅 ∧ ∀𝑧 ∈ (ℝ × ℝ)∀𝑤 ∈ (ℝ × ℝ)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
7225, 70, 71mpbir2an 711 . 2 𝐹:(ℝ × ℝ)–1-1𝑅
7319eleq2i 2831 . . . . . . . 8 (𝑤𝑅𝑤 ∈ (ℝ ↑m {1, 2}))
74 reex 11244 . . . . . . . . 9 ℝ ∈ V
75 prex 5443 . . . . . . . . 9 {1, 2} ∈ V
7674, 75elmap 8910 . . . . . . . 8 (𝑤 ∈ (ℝ ↑m {1, 2}) ↔ 𝑤:{1, 2}⟶ℝ)
77 1re 11259 . . . . . . . . 9 1 ∈ ℝ
78 2re 12338 . . . . . . . . 9 2 ∈ ℝ
79 fpr2g 7231 . . . . . . . . 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 692 . . . . . . . 8 (𝑤:{1, 2}⟶ℝ ↔ ((𝑤‘1) ∈ ℝ ∧ (𝑤‘2) ∈ ℝ ∧ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩}))
8173, 76, 803bitri 297 . . . . . . 7 (𝑤𝑅 ↔ ((𝑤‘1) ∈ ℝ ∧ (𝑤‘2) ∈ ℝ ∧ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩}))
82 opeq2 4879 . . . . . . . . . 10 (𝑢 = (𝑤‘1) → ⟨1, 𝑢⟩ = ⟨1, (𝑤‘1)⟩)
8382preq1d 4744 . . . . . . . . 9 (𝑢 = (𝑤‘1) → {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩})
8483eqeq2d 2746 . . . . . . . 8 (𝑢 = (𝑤‘1) → (𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} ↔ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩}))
85 opeq2 4879 . . . . . . . . . 10 (𝑣 = (𝑤‘2) → ⟨2, 𝑣⟩ = ⟨2, (𝑤‘2)⟩)
8685preq2d 4745 . . . . . . . . 9 (𝑣 = (𝑤‘2) → {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩} = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩})
8786eqeq2d 2746 . . . . . . . 8 (𝑣 = (𝑤‘2) → (𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩} ↔ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩}))
8884, 87rspc2ev 3635 . . . . . . 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 217 . . . . . 6 (𝑤𝑅 → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
90 opeq2 4879 . . . . . . . . . 10 (𝑥 = 𝑢 → ⟨1, 𝑥⟩ = ⟨1, 𝑢⟩)
9190preq1d 4744 . . . . . . . . 9 (𝑥 = 𝑢 → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑢⟩, ⟨2, 𝑦⟩})
92 opeq2 4879 . . . . . . . . . 10 (𝑦 = 𝑣 → ⟨2, 𝑦⟩ = ⟨2, 𝑣⟩)
9392preq2d 4745 . . . . . . . . 9 (𝑦 = 𝑣 → {⟨1, 𝑢⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
94 prex 5443 . . . . . . . . 9 {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} ∈ V
9591, 93, 1, 94ovmpo 7593 . . . . . . . 8 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢𝐹𝑣) = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
9695eqeq2d 2746 . . . . . . 7 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑤 = (𝑢𝐹𝑣) ↔ 𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩}))
97962rexbiia 3216 . . . . . 6 (∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
9889, 97sylibr 234 . . . . 5 (𝑤𝑅 → ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
99 fveq2 6907 . . . . . . . 8 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹𝑧) = (𝐹‘⟨𝑢, 𝑣⟩))
100 df-ov 7434 . . . . . . . 8 (𝑢𝐹𝑣) = (𝐹‘⟨𝑢, 𝑣⟩)
10199, 100eqtr4di 2793 . . . . . . 7 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹𝑧) = (𝑢𝐹𝑣))
102101eqeq2d 2746 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝑤 = (𝐹𝑧) ↔ 𝑤 = (𝑢𝐹𝑣)))
103102rexxp 5856 . . . . 5 (∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
10498, 103sylibr 234 . . . 4 (𝑤𝑅 → ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧))
105104rgen 3061 . . 3 𝑤𝑅𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧)
106 dffo3 7122 . . 3 (𝐹:(ℝ × ℝ)–onto𝑅 ↔ (𝐹:(ℝ × ℝ)⟶𝑅 ∧ ∀𝑤𝑅𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧)))
10725, 105, 106mpbir2an 711 . 2 𝐹:(ℝ × ℝ)–onto𝑅
108 df-f1o 6570 . 2 (𝐹:(ℝ × ℝ)–1-1-onto𝑅 ↔ (𝐹:(ℝ × ℝ)–1-1𝑅𝐹:(ℝ × ℝ)–onto𝑅))
10972, 107, 108mpbir2an 711 1 𝐹:(ℝ × ℝ)–1-1-onto𝑅
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  {cpr 4633  cop 4637   × cxp 5687   Fn wfn 6558  wf 6559  1-1wf1 6560  ontowfo 6561  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012  m cmap 8865  cr 11152  1c1 11154  2c2 12319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-2 12327
This theorem is referenced by:  rrx2xpreen  48569  rrx2plordisom  48573
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