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Theorem rrx2xpref1o 48639
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 5437 . . . . 5 {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} ∈ V
31, 2fnmpoi 8095 . . . 4 𝐹 Fn (ℝ × ℝ)
4 1st2nd2 8053 . . . . . . . . 9 (𝑧 ∈ (ℝ × ℝ) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
54fveq2d 6910 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩))
6 df-ov 7434 . . . . . . . 8 ((1st𝑧)𝐹(2nd𝑧)) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩)
75, 6eqtr4di 2795 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = ((1st𝑧)𝐹(2nd𝑧)))
8 xp1st 8046 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℝ)
9 xp2nd 8047 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (2nd𝑧) ∈ ℝ)
10 opeq2 4874 . . . . . . . . . 10 (𝑥 = (1st𝑧) → ⟨1, 𝑥⟩ = ⟨1, (1st𝑧)⟩)
1110preq1d 4739 . . . . . . . . 9 (𝑥 = (1st𝑧) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, (1st𝑧)⟩, ⟨2, 𝑦⟩})
12 opeq2 4874 . . . . . . . . . 10 (𝑦 = (2nd𝑧) → ⟨2, 𝑦⟩ = ⟨2, (2nd𝑧)⟩)
1312preq2d 4740 . . . . . . . . 9 (𝑦 = (2nd𝑧) → {⟨1, (1st𝑧)⟩, ⟨2, 𝑦⟩} = {⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩})
14 prex 5437 . . . . . . . . 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 2777 . . . . . 6 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) = {⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩})
18 eqid 2737 . . . . . . . 8 {1, 2} = {1, 2}
19 rrx2xpreen.r . . . . . . . 8 𝑅 = (ℝ ↑m {1, 2})
2018, 19prelrrx2 48634 . . . . . . 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 2841 . . . . 5 (𝑧 ∈ (ℝ × ℝ) → (𝐹𝑧) ∈ 𝑅)
2322rgen 3063 . . . 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 5469 . . . . . . . 8 ⟨1, (1st𝑧)⟩ ∈ V
27 opex 5469 . . . . . . . 8 ⟨2, (2nd𝑧)⟩ ∈ V
28 opex 5469 . . . . . . . 8 ⟨1, (1st𝑤)⟩ ∈ V
29 opex 5469 . . . . . . . 8 ⟨2, (2nd𝑤)⟩ ∈ V
3026, 27, 28, 29preq12b 4850 . . . . . . 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 11257 . . . . . . . . . . . 12 1 ∈ V
32 fvex 6919 . . . . . . . . . . . 12 (1st𝑧) ∈ V
3331, 32opth 5481 . . . . . . . . . . 11 (⟨1, (1st𝑧)⟩ = ⟨1, (1st𝑤)⟩ ↔ (1 = 1 ∧ (1st𝑧) = (1st𝑤)))
3433simprbi 496 . . . . . . . . . 10 (⟨1, (1st𝑧)⟩ = ⟨1, (1st𝑤)⟩ → (1st𝑧) = (1st𝑤))
35 2ex 12343 . . . . . . . . . . . 12 2 ∈ V
36 fvex 6919 . . . . . . . . . . . 12 (2nd𝑧) ∈ V
3735, 36opth 5481 . . . . . . . . . . 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 5481 . . . . . . . . 9 (⟨1, (1st𝑧)⟩ = ⟨2, (2nd𝑤)⟩ ↔ (1 = 2 ∧ (1st𝑧) = (2nd𝑤)))
4235, 36opth 5481 . . . . . . . . 9 (⟨2, (2nd𝑧)⟩ = ⟨1, (1st𝑤)⟩ ↔ (2 = 1 ∧ (2nd𝑧) = (1st𝑤)))
43 1ne2 12474 . . . . . . . . . . 11 1 ≠ 2
44 eqneqall 2951 . . . . . . . . . . 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 858 . . . . . . 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 8053 . . . . . . . . 9 (𝑤 ∈ (ℝ × ℝ) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
5251fveq2d 6910 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → (𝐹𝑤) = (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩))
53 df-ov 7434 . . . . . . . 8 ((1st𝑤)𝐹(2nd𝑤)) = (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩)
5452, 53eqtr4di 2795 . . . . . . 7 (𝑤 ∈ (ℝ × ℝ) → (𝐹𝑤) = ((1st𝑤)𝐹(2nd𝑤)))
55 xp1st 8046 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → (1st𝑤) ∈ ℝ)
56 xp2nd 8047 . . . . . . . 8 (𝑤 ∈ (ℝ × ℝ) → (2nd𝑤) ∈ ℝ)
57 opeq2 4874 . . . . . . . . . 10 (𝑥 = (1st𝑤) → ⟨1, 𝑥⟩ = ⟨1, (1st𝑤)⟩)
5857preq1d 4739 . . . . . . . . 9 (𝑥 = (1st𝑤) → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, (1st𝑤)⟩, ⟨2, 𝑦⟩})
59 opeq2 4874 . . . . . . . . . 10 (𝑦 = (2nd𝑤) → ⟨2, 𝑦⟩ = ⟨2, (2nd𝑤)⟩)
6059preq2d 4740 . . . . . . . . 9 (𝑦 = (2nd𝑤) → {⟨1, (1st𝑤)⟩, ⟨2, 𝑦⟩} = {⟨1, (1st𝑤)⟩, ⟨2, (2nd𝑤)⟩})
61 prex 5437 . . . . . . . . 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 2777 . . . . . 6 (𝑤 ∈ (ℝ × ℝ) → (𝐹𝑤) = {⟨1, (1st𝑤)⟩, ⟨2, (2nd𝑤)⟩})
6517, 64eqeqan12d 2751 . . . . 5 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → ((𝐹𝑧) = (𝐹𝑤) ↔ {⟨1, (1st𝑧)⟩, ⟨2, (2nd𝑧)⟩} = {⟨1, (1st𝑤)⟩, ⟨2, (2nd𝑤)⟩}))
664, 51eqeqan12d 2751 . . . . . 6 ((𝑧 ∈ (ℝ × ℝ) ∧ 𝑤 ∈ (ℝ × ℝ)) → (𝑧 = 𝑤 ↔ ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩))
6732, 36opth 5481 . . . . . 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 3199 . . 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 2833 . . . . . . . 8 (𝑤𝑅𝑤 ∈ (ℝ ↑m {1, 2}))
74 reex 11246 . . . . . . . . 9 ℝ ∈ V
75 prex 5437 . . . . . . . . 9 {1, 2} ∈ V
7674, 75elmap 8911 . . . . . . . 8 (𝑤 ∈ (ℝ ↑m {1, 2}) ↔ 𝑤:{1, 2}⟶ℝ)
77 1re 11261 . . . . . . . . 9 1 ∈ ℝ
78 2re 12340 . . . . . . . . 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 4874 . . . . . . . . . 10 (𝑢 = (𝑤‘1) → ⟨1, 𝑢⟩ = ⟨1, (𝑤‘1)⟩)
8382preq1d 4739 . . . . . . . . 9 (𝑢 = (𝑤‘1) → {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩})
8483eqeq2d 2748 . . . . . . . 8 (𝑢 = (𝑤‘1) → (𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} ↔ 𝑤 = {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩}))
85 opeq2 4874 . . . . . . . . . 10 (𝑣 = (𝑤‘2) → ⟨2, 𝑣⟩ = ⟨2, (𝑤‘2)⟩)
8685preq2d 4740 . . . . . . . . 9 (𝑣 = (𝑤‘2) → {⟨1, (𝑤‘1)⟩, ⟨2, 𝑣⟩} = {⟨1, (𝑤‘1)⟩, ⟨2, (𝑤‘2)⟩})
8786eqeq2d 2748 . . . . . . . 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 4874 . . . . . . . . . 10 (𝑥 = 𝑢 → ⟨1, 𝑥⟩ = ⟨1, 𝑢⟩)
9190preq1d 4739 . . . . . . . . 9 (𝑥 = 𝑢 → {⟨1, 𝑥⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑢⟩, ⟨2, 𝑦⟩})
92 opeq2 4874 . . . . . . . . . 10 (𝑦 = 𝑣 → ⟨2, 𝑦⟩ = ⟨2, 𝑣⟩)
9392preq2d 4740 . . . . . . . . 9 (𝑦 = 𝑣 → {⟨1, 𝑢⟩, ⟨2, 𝑦⟩} = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
94 prex 5437 . . . . . . . . 9 {⟨1, 𝑢⟩, ⟨2, 𝑣⟩} ∈ V
9591, 93, 1, 94ovmpo 7593 . . . . . . . 8 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢𝐹𝑣) = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩})
9695eqeq2d 2748 . . . . . . 7 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑤 = (𝑢𝐹𝑣) ↔ 𝑤 = {⟨1, 𝑢⟩, ⟨2, 𝑣⟩}))
97962rexbiia 3218 . . . . . 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 6906 . . . . . . . 8 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹𝑧) = (𝐹‘⟨𝑢, 𝑣⟩))
100 df-ov 7434 . . . . . . . 8 (𝑢𝐹𝑣) = (𝐹‘⟨𝑢, 𝑣⟩)
10199, 100eqtr4di 2795 . . . . . . 7 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝐹𝑧) = (𝑢𝐹𝑣))
102101eqeq2d 2748 . . . . . 6 (𝑧 = ⟨𝑢, 𝑣⟩ → (𝑤 = (𝐹𝑧) ↔ 𝑤 = (𝑢𝐹𝑣)))
103102rexxp 5853 . . . . 5 (∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧) ↔ ∃𝑢 ∈ ℝ ∃𝑣 ∈ ℝ 𝑤 = (𝑢𝐹𝑣))
10498, 103sylibr 234 . . . 4 (𝑤𝑅 → ∃𝑧 ∈ (ℝ × ℝ)𝑤 = (𝐹𝑧))
105104rgen 3063 . . 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 6568 . 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 848  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  {cpr 4628  cop 4632   × cxp 5683   Fn wfn 6556  wf 6557  1-1wf1 6558  ontowfo 6559  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  2nd c2nd 8013  m cmap 8866  cr 11154  1c1 11156  2c2 12321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-2 12329
This theorem is referenced by:  rrx2xpreen  48640  rrx2plordisom  48644
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