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Theorem rrx2xpref1o 47404
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 = (π‘₯ ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, π‘₯⟩, ⟨2, π‘¦βŸ©})
Assertion
Ref Expression
rrx2xpref1o 𝐹:(ℝ Γ— ℝ)–1-1-onto→𝑅
Distinct variable group:   π‘₯,𝑦
Allowed substitution hints:   𝑅(π‘₯,𝑦)   𝐹(π‘₯,𝑦)

Proof of Theorem rrx2xpref1o
Dummy variables 𝑣 𝑒 𝑀 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rrx2xpref1o.1 . . . . 5 𝐹 = (π‘₯ ∈ ℝ, 𝑦 ∈ ℝ ↦ {⟨1, π‘₯⟩, ⟨2, π‘¦βŸ©})
2 prex 5433 . . . . 5 {⟨1, π‘₯⟩, ⟨2, π‘¦βŸ©} ∈ V
31, 2fnmpoi 8056 . . . 4 𝐹 Fn (ℝ Γ— ℝ)
4 1st2nd2 8014 . . . . . . . . 9 (𝑧 ∈ (ℝ Γ— ℝ) β†’ 𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
54fveq2d 6896 . . . . . . . 8 (𝑧 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘§) = (πΉβ€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩))
6 df-ov 7412 . . . . . . . 8 ((1st β€˜π‘§)𝐹(2nd β€˜π‘§)) = (πΉβ€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
75, 6eqtr4di 2791 . . . . . . 7 (𝑧 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘§) = ((1st β€˜π‘§)𝐹(2nd β€˜π‘§)))
8 xp1st 8007 . . . . . . . 8 (𝑧 ∈ (ℝ Γ— ℝ) β†’ (1st β€˜π‘§) ∈ ℝ)
9 xp2nd 8008 . . . . . . . 8 (𝑧 ∈ (ℝ Γ— ℝ) β†’ (2nd β€˜π‘§) ∈ ℝ)
10 opeq2 4875 . . . . . . . . . 10 (π‘₯ = (1st β€˜π‘§) β†’ ⟨1, π‘₯⟩ = ⟨1, (1st β€˜π‘§)⟩)
1110preq1d 4744 . . . . . . . . 9 (π‘₯ = (1st β€˜π‘§) β†’ {⟨1, π‘₯⟩, ⟨2, π‘¦βŸ©} = {⟨1, (1st β€˜π‘§)⟩, ⟨2, π‘¦βŸ©})
12 opeq2 4875 . . . . . . . . . 10 (𝑦 = (2nd β€˜π‘§) β†’ ⟨2, π‘¦βŸ© = ⟨2, (2nd β€˜π‘§)⟩)
1312preq2d 4745 . . . . . . . . 9 (𝑦 = (2nd β€˜π‘§) β†’ {⟨1, (1st β€˜π‘§)⟩, ⟨2, π‘¦βŸ©} = {⟨1, (1st β€˜π‘§)⟩, ⟨2, (2nd β€˜π‘§)⟩})
14 prex 5433 . . . . . . . . 9 {⟨1, (1st β€˜π‘§)⟩, ⟨2, (2nd β€˜π‘§)⟩} ∈ V
1511, 13, 1, 14ovmpo 7568 . . . . . . . 8 (((1st β€˜π‘§) ∈ ℝ ∧ (2nd β€˜π‘§) ∈ ℝ) β†’ ((1st β€˜π‘§)𝐹(2nd β€˜π‘§)) = {⟨1, (1st β€˜π‘§)⟩, ⟨2, (2nd β€˜π‘§)⟩})
168, 9, 15syl2anc 585 . . . . . . 7 (𝑧 ∈ (ℝ Γ— ℝ) β†’ ((1st β€˜π‘§)𝐹(2nd β€˜π‘§)) = {⟨1, (1st β€˜π‘§)⟩, ⟨2, (2nd β€˜π‘§)⟩})
177, 16eqtrd 2773 . . . . . 6 (𝑧 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘§) = {⟨1, (1st β€˜π‘§)⟩, ⟨2, (2nd β€˜π‘§)⟩})
18 eqid 2733 . . . . . . . 8 {1, 2} = {1, 2}
19 rrx2xpreen.r . . . . . . . 8 𝑅 = (ℝ ↑m {1, 2})
2018, 19prelrrx2 47399 . . . . . . 7 (((1st β€˜π‘§) ∈ ℝ ∧ (2nd β€˜π‘§) ∈ ℝ) β†’ {⟨1, (1st β€˜π‘§)⟩, ⟨2, (2nd β€˜π‘§)⟩} ∈ 𝑅)
218, 9, 20syl2anc 585 . . . . . 6 (𝑧 ∈ (ℝ Γ— ℝ) β†’ {⟨1, (1st β€˜π‘§)⟩, ⟨2, (2nd β€˜π‘§)⟩} ∈ 𝑅)
2217, 21eqeltrd 2834 . . . . 5 (𝑧 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘§) ∈ 𝑅)
2322rgen 3064 . . . 4 βˆ€π‘§ ∈ (ℝ Γ— ℝ)(πΉβ€˜π‘§) ∈ 𝑅
24 ffnfv 7118 . . . 4 (𝐹:(ℝ Γ— ℝ)βŸΆπ‘… ↔ (𝐹 Fn (ℝ Γ— ℝ) ∧ βˆ€π‘§ ∈ (ℝ Γ— ℝ)(πΉβ€˜π‘§) ∈ 𝑅))
253, 23, 24mpbir2an 710 . . 3 𝐹:(ℝ Γ— ℝ)βŸΆπ‘…
26 opex 5465 . . . . . . . 8 ⟨1, (1st β€˜π‘§)⟩ ∈ V
27 opex 5465 . . . . . . . 8 ⟨2, (2nd β€˜π‘§)⟩ ∈ V
28 opex 5465 . . . . . . . 8 ⟨1, (1st β€˜π‘€)⟩ ∈ V
29 opex 5465 . . . . . . . 8 ⟨2, (2nd β€˜π‘€)⟩ ∈ V
3026, 27, 28, 29preq12b 4852 . . . . . . 7 ({⟨1, (1st β€˜π‘§)⟩, ⟨2, (2nd β€˜π‘§)⟩} = {⟨1, (1st β€˜π‘€)⟩, ⟨2, (2nd β€˜π‘€)⟩} ↔ ((⟨1, (1st β€˜π‘§)⟩ = ⟨1, (1st β€˜π‘€)⟩ ∧ ⟨2, (2nd β€˜π‘§)⟩ = ⟨2, (2nd β€˜π‘€)⟩) ∨ (⟨1, (1st β€˜π‘§)⟩ = ⟨2, (2nd β€˜π‘€)⟩ ∧ ⟨2, (2nd β€˜π‘§)⟩ = ⟨1, (1st β€˜π‘€)⟩)))
31 1ex 11210 . . . . . . . . . . . 12 1 ∈ V
32 fvex 6905 . . . . . . . . . . . 12 (1st β€˜π‘§) ∈ V
3331, 32opth 5477 . . . . . . . . . . 11 (⟨1, (1st β€˜π‘§)⟩ = ⟨1, (1st β€˜π‘€)⟩ ↔ (1 = 1 ∧ (1st β€˜π‘§) = (1st β€˜π‘€)))
3433simprbi 498 . . . . . . . . . 10 (⟨1, (1st β€˜π‘§)⟩ = ⟨1, (1st β€˜π‘€)⟩ β†’ (1st β€˜π‘§) = (1st β€˜π‘€))
35 2ex 12289 . . . . . . . . . . . 12 2 ∈ V
36 fvex 6905 . . . . . . . . . . . 12 (2nd β€˜π‘§) ∈ V
3735, 36opth 5477 . . . . . . . . . . 11 (⟨2, (2nd β€˜π‘§)⟩ = ⟨2, (2nd β€˜π‘€)⟩ ↔ (2 = 2 ∧ (2nd β€˜π‘§) = (2nd β€˜π‘€)))
3837simprbi 498 . . . . . . . . . 10 (⟨2, (2nd β€˜π‘§)⟩ = ⟨2, (2nd β€˜π‘€)⟩ β†’ (2nd β€˜π‘§) = (2nd β€˜π‘€))
3934, 38anim12i 614 . . . . . . . . 9 ((⟨1, (1st β€˜π‘§)⟩ = ⟨1, (1st β€˜π‘€)⟩ ∧ ⟨2, (2nd β€˜π‘§)⟩ = ⟨2, (2nd β€˜π‘€)⟩) β†’ ((1st β€˜π‘§) = (1st β€˜π‘€) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘€)))
4039a1d 25 . . . . . . . 8 ((⟨1, (1st β€˜π‘§)⟩ = ⟨1, (1st β€˜π‘€)⟩ ∧ ⟨2, (2nd β€˜π‘§)⟩ = ⟨2, (2nd β€˜π‘€)⟩) β†’ ((𝑧 ∈ (ℝ Γ— ℝ) ∧ 𝑀 ∈ (ℝ Γ— ℝ)) β†’ ((1st β€˜π‘§) = (1st β€˜π‘€) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘€))))
4131, 32opth 5477 . . . . . . . . 9 (⟨1, (1st β€˜π‘§)⟩ = ⟨2, (2nd β€˜π‘€)⟩ ↔ (1 = 2 ∧ (1st β€˜π‘§) = (2nd β€˜π‘€)))
4235, 36opth 5477 . . . . . . . . 9 (⟨2, (2nd β€˜π‘§)⟩ = ⟨1, (1st β€˜π‘€)⟩ ↔ (2 = 1 ∧ (2nd β€˜π‘§) = (1st β€˜π‘€)))
43 1ne2 12420 . . . . . . . . . . 11 1 β‰  2
44 eqneqall 2952 . . . . . . . . . . 11 (1 = 2 β†’ (1 β‰  2 β†’ ((𝑧 ∈ (ℝ Γ— ℝ) ∧ 𝑀 ∈ (ℝ Γ— ℝ)) β†’ ((1st β€˜π‘§) = (1st β€˜π‘€) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘€)))))
4543, 44mpi 20 . . . . . . . . . 10 (1 = 2 β†’ ((𝑧 ∈ (ℝ Γ— ℝ) ∧ 𝑀 ∈ (ℝ Γ— ℝ)) β†’ ((1st β€˜π‘§) = (1st β€˜π‘€) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘€))))
4645ad2antrr 725 . . . . . . . . 9 (((1 = 2 ∧ (1st β€˜π‘§) = (2nd β€˜π‘€)) ∧ (2 = 1 ∧ (2nd β€˜π‘§) = (1st β€˜π‘€))) β†’ ((𝑧 ∈ (ℝ Γ— ℝ) ∧ 𝑀 ∈ (ℝ Γ— ℝ)) β†’ ((1st β€˜π‘§) = (1st β€˜π‘€) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘€))))
4741, 42, 46syl2anb 599 . . . . . . . 8 ((⟨1, (1st β€˜π‘§)⟩ = ⟨2, (2nd β€˜π‘€)⟩ ∧ ⟨2, (2nd β€˜π‘§)⟩ = ⟨1, (1st β€˜π‘€)⟩) β†’ ((𝑧 ∈ (ℝ Γ— ℝ) ∧ 𝑀 ∈ (ℝ Γ— ℝ)) β†’ ((1st β€˜π‘§) = (1st β€˜π‘€) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘€))))
4840, 47jaoi 856 . . . . . . 7 (((⟨1, (1st β€˜π‘§)⟩ = ⟨1, (1st β€˜π‘€)⟩ ∧ ⟨2, (2nd β€˜π‘§)⟩ = ⟨2, (2nd β€˜π‘€)⟩) ∨ (⟨1, (1st β€˜π‘§)⟩ = ⟨2, (2nd β€˜π‘€)⟩ ∧ ⟨2, (2nd β€˜π‘§)⟩ = ⟨1, (1st β€˜π‘€)⟩)) β†’ ((𝑧 ∈ (ℝ Γ— ℝ) ∧ 𝑀 ∈ (ℝ Γ— ℝ)) β†’ ((1st β€˜π‘§) = (1st β€˜π‘€) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘€))))
4930, 48sylbi 216 . . . . . 6 ({⟨1, (1st β€˜π‘§)⟩, ⟨2, (2nd β€˜π‘§)⟩} = {⟨1, (1st β€˜π‘€)⟩, ⟨2, (2nd β€˜π‘€)⟩} β†’ ((𝑧 ∈ (ℝ Γ— ℝ) ∧ 𝑀 ∈ (ℝ Γ— ℝ)) β†’ ((1st β€˜π‘§) = (1st β€˜π‘€) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘€))))
5049com12 32 . . . . 5 ((𝑧 ∈ (ℝ Γ— ℝ) ∧ 𝑀 ∈ (ℝ Γ— ℝ)) β†’ ({⟨1, (1st β€˜π‘§)⟩, ⟨2, (2nd β€˜π‘§)⟩} = {⟨1, (1st β€˜π‘€)⟩, ⟨2, (2nd β€˜π‘€)⟩} β†’ ((1st β€˜π‘§) = (1st β€˜π‘€) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘€))))
51 1st2nd2 8014 . . . . . . . . 9 (𝑀 ∈ (ℝ Γ— ℝ) β†’ 𝑀 = ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)
5251fveq2d 6896 . . . . . . . 8 (𝑀 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘€) = (πΉβ€˜βŸ¨(1st β€˜π‘€), (2nd β€˜π‘€)⟩))
53 df-ov 7412 . . . . . . . 8 ((1st β€˜π‘€)𝐹(2nd β€˜π‘€)) = (πΉβ€˜βŸ¨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)
5452, 53eqtr4di 2791 . . . . . . 7 (𝑀 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘€) = ((1st β€˜π‘€)𝐹(2nd β€˜π‘€)))
55 xp1st 8007 . . . . . . . 8 (𝑀 ∈ (ℝ Γ— ℝ) β†’ (1st β€˜π‘€) ∈ ℝ)
56 xp2nd 8008 . . . . . . . 8 (𝑀 ∈ (ℝ Γ— ℝ) β†’ (2nd β€˜π‘€) ∈ ℝ)
57 opeq2 4875 . . . . . . . . . 10 (π‘₯ = (1st β€˜π‘€) β†’ ⟨1, π‘₯⟩ = ⟨1, (1st β€˜π‘€)⟩)
5857preq1d 4744 . . . . . . . . 9 (π‘₯ = (1st β€˜π‘€) β†’ {⟨1, π‘₯⟩, ⟨2, π‘¦βŸ©} = {⟨1, (1st β€˜π‘€)⟩, ⟨2, π‘¦βŸ©})
59 opeq2 4875 . . . . . . . . . 10 (𝑦 = (2nd β€˜π‘€) β†’ ⟨2, π‘¦βŸ© = ⟨2, (2nd β€˜π‘€)⟩)
6059preq2d 4745 . . . . . . . . 9 (𝑦 = (2nd β€˜π‘€) β†’ {⟨1, (1st β€˜π‘€)⟩, ⟨2, π‘¦βŸ©} = {⟨1, (1st β€˜π‘€)⟩, ⟨2, (2nd β€˜π‘€)⟩})
61 prex 5433 . . . . . . . . 9 {⟨1, (1st β€˜π‘€)⟩, ⟨2, (2nd β€˜π‘€)⟩} ∈ V
6258, 60, 1, 61ovmpo 7568 . . . . . . . 8 (((1st β€˜π‘€) ∈ ℝ ∧ (2nd β€˜π‘€) ∈ ℝ) β†’ ((1st β€˜π‘€)𝐹(2nd β€˜π‘€)) = {⟨1, (1st β€˜π‘€)⟩, ⟨2, (2nd β€˜π‘€)⟩})
6355, 56, 62syl2anc 585 . . . . . . 7 (𝑀 ∈ (ℝ Γ— ℝ) β†’ ((1st β€˜π‘€)𝐹(2nd β€˜π‘€)) = {⟨1, (1st β€˜π‘€)⟩, ⟨2, (2nd β€˜π‘€)⟩})
6454, 63eqtrd 2773 . . . . . 6 (𝑀 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘€) = {⟨1, (1st β€˜π‘€)⟩, ⟨2, (2nd β€˜π‘€)⟩})
6517, 64eqeqan12d 2747 . . . . 5 ((𝑧 ∈ (ℝ Γ— ℝ) ∧ 𝑀 ∈ (ℝ Γ— ℝ)) β†’ ((πΉβ€˜π‘§) = (πΉβ€˜π‘€) ↔ {⟨1, (1st β€˜π‘§)⟩, ⟨2, (2nd β€˜π‘§)⟩} = {⟨1, (1st β€˜π‘€)⟩, ⟨2, (2nd β€˜π‘€)⟩}))
664, 51eqeqan12d 2747 . . . . . 6 ((𝑧 ∈ (ℝ Γ— ℝ) ∧ 𝑀 ∈ (ℝ Γ— ℝ)) β†’ (𝑧 = 𝑀 ↔ ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩ = ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩))
6732, 36opth 5477 . . . . . 6 (⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩ = ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩ ↔ ((1st β€˜π‘§) = (1st β€˜π‘€) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘€)))
6866, 67bitrdi 287 . . . . 5 ((𝑧 ∈ (ℝ Γ— ℝ) ∧ 𝑀 ∈ (ℝ Γ— ℝ)) β†’ (𝑧 = 𝑀 ↔ ((1st β€˜π‘§) = (1st β€˜π‘€) ∧ (2nd β€˜π‘§) = (2nd β€˜π‘€))))
6950, 65, 683imtr4d 294 . . . 4 ((𝑧 ∈ (ℝ Γ— ℝ) ∧ 𝑀 ∈ (ℝ Γ— ℝ)) β†’ ((πΉβ€˜π‘§) = (πΉβ€˜π‘€) β†’ 𝑧 = 𝑀))
7069rgen2 3198 . . 3 βˆ€π‘§ ∈ (ℝ Γ— ℝ)βˆ€π‘€ ∈ (ℝ Γ— ℝ)((πΉβ€˜π‘§) = (πΉβ€˜π‘€) β†’ 𝑧 = 𝑀)
71 dff13 7254 . . 3 (𝐹:(ℝ Γ— ℝ)–1-1→𝑅 ↔ (𝐹:(ℝ Γ— ℝ)βŸΆπ‘… ∧ βˆ€π‘§ ∈ (ℝ Γ— ℝ)βˆ€π‘€ ∈ (ℝ Γ— ℝ)((πΉβ€˜π‘§) = (πΉβ€˜π‘€) β†’ 𝑧 = 𝑀)))
7225, 70, 71mpbir2an 710 . 2 𝐹:(ℝ Γ— ℝ)–1-1→𝑅
7319eleq2i 2826 . . . . . . . 8 (𝑀 ∈ 𝑅 ↔ 𝑀 ∈ (ℝ ↑m {1, 2}))
74 reex 11201 . . . . . . . . 9 ℝ ∈ V
75 prex 5433 . . . . . . . . 9 {1, 2} ∈ V
7674, 75elmap 8865 . . . . . . . 8 (𝑀 ∈ (ℝ ↑m {1, 2}) ↔ 𝑀:{1, 2}βŸΆβ„)
77 1re 11214 . . . . . . . . 9 1 ∈ ℝ
78 2re 12286 . . . . . . . . 9 2 ∈ ℝ
79 fpr2g 7213 . . . . . . . . 9 ((1 ∈ ℝ ∧ 2 ∈ ℝ) β†’ (𝑀:{1, 2}βŸΆβ„ ↔ ((π‘€β€˜1) ∈ ℝ ∧ (π‘€β€˜2) ∈ ℝ ∧ 𝑀 = {⟨1, (π‘€β€˜1)⟩, ⟨2, (π‘€β€˜2)⟩})))
8077, 78, 79mp2an 691 . . . . . . . 8 (𝑀:{1, 2}βŸΆβ„ ↔ ((π‘€β€˜1) ∈ ℝ ∧ (π‘€β€˜2) ∈ ℝ ∧ 𝑀 = {⟨1, (π‘€β€˜1)⟩, ⟨2, (π‘€β€˜2)⟩}))
8173, 76, 803bitri 297 . . . . . . 7 (𝑀 ∈ 𝑅 ↔ ((π‘€β€˜1) ∈ ℝ ∧ (π‘€β€˜2) ∈ ℝ ∧ 𝑀 = {⟨1, (π‘€β€˜1)⟩, ⟨2, (π‘€β€˜2)⟩}))
82 opeq2 4875 . . . . . . . . . 10 (𝑒 = (π‘€β€˜1) β†’ ⟨1, π‘’βŸ© = ⟨1, (π‘€β€˜1)⟩)
8382preq1d 4744 . . . . . . . . 9 (𝑒 = (π‘€β€˜1) β†’ {⟨1, π‘’βŸ©, ⟨2, π‘£βŸ©} = {⟨1, (π‘€β€˜1)⟩, ⟨2, π‘£βŸ©})
8483eqeq2d 2744 . . . . . . . 8 (𝑒 = (π‘€β€˜1) β†’ (𝑀 = {⟨1, π‘’βŸ©, ⟨2, π‘£βŸ©} ↔ 𝑀 = {⟨1, (π‘€β€˜1)⟩, ⟨2, π‘£βŸ©}))
85 opeq2 4875 . . . . . . . . . 10 (𝑣 = (π‘€β€˜2) β†’ ⟨2, π‘£βŸ© = ⟨2, (π‘€β€˜2)⟩)
8685preq2d 4745 . . . . . . . . 9 (𝑣 = (π‘€β€˜2) β†’ {⟨1, (π‘€β€˜1)⟩, ⟨2, π‘£βŸ©} = {⟨1, (π‘€β€˜1)⟩, ⟨2, (π‘€β€˜2)⟩})
8786eqeq2d 2744 . . . . . . . 8 (𝑣 = (π‘€β€˜2) β†’ (𝑀 = {⟨1, (π‘€β€˜1)⟩, ⟨2, π‘£βŸ©} ↔ 𝑀 = {⟨1, (π‘€β€˜1)⟩, ⟨2, (π‘€β€˜2)⟩}))
8884, 87rspc2ev 3625 . . . . . . 7 (((π‘€β€˜1) ∈ ℝ ∧ (π‘€β€˜2) ∈ ℝ ∧ 𝑀 = {⟨1, (π‘€β€˜1)⟩, ⟨2, (π‘€β€˜2)⟩}) β†’ βˆƒπ‘’ ∈ ℝ βˆƒπ‘£ ∈ ℝ 𝑀 = {⟨1, π‘’βŸ©, ⟨2, π‘£βŸ©})
8981, 88sylbi 216 . . . . . 6 (𝑀 ∈ 𝑅 β†’ βˆƒπ‘’ ∈ ℝ βˆƒπ‘£ ∈ ℝ 𝑀 = {⟨1, π‘’βŸ©, ⟨2, π‘£βŸ©})
90 opeq2 4875 . . . . . . . . . 10 (π‘₯ = 𝑒 β†’ ⟨1, π‘₯⟩ = ⟨1, π‘’βŸ©)
9190preq1d 4744 . . . . . . . . 9 (π‘₯ = 𝑒 β†’ {⟨1, π‘₯⟩, ⟨2, π‘¦βŸ©} = {⟨1, π‘’βŸ©, ⟨2, π‘¦βŸ©})
92 opeq2 4875 . . . . . . . . . 10 (𝑦 = 𝑣 β†’ ⟨2, π‘¦βŸ© = ⟨2, π‘£βŸ©)
9392preq2d 4745 . . . . . . . . 9 (𝑦 = 𝑣 β†’ {⟨1, π‘’βŸ©, ⟨2, π‘¦βŸ©} = {⟨1, π‘’βŸ©, ⟨2, π‘£βŸ©})
94 prex 5433 . . . . . . . . 9 {⟨1, π‘’βŸ©, ⟨2, π‘£βŸ©} ∈ V
9591, 93, 1, 94ovmpo 7568 . . . . . . . 8 ((𝑒 ∈ ℝ ∧ 𝑣 ∈ ℝ) β†’ (𝑒𝐹𝑣) = {⟨1, π‘’βŸ©, ⟨2, π‘£βŸ©})
9695eqeq2d 2744 . . . . . . 7 ((𝑒 ∈ ℝ ∧ 𝑣 ∈ ℝ) β†’ (𝑀 = (𝑒𝐹𝑣) ↔ 𝑀 = {⟨1, π‘’βŸ©, ⟨2, π‘£βŸ©}))
97962rexbiia 3216 . . . . . 6 (βˆƒπ‘’ ∈ ℝ βˆƒπ‘£ ∈ ℝ 𝑀 = (𝑒𝐹𝑣) ↔ βˆƒπ‘’ ∈ ℝ βˆƒπ‘£ ∈ ℝ 𝑀 = {⟨1, π‘’βŸ©, ⟨2, π‘£βŸ©})
9889, 97sylibr 233 . . . . 5 (𝑀 ∈ 𝑅 β†’ βˆƒπ‘’ ∈ ℝ βˆƒπ‘£ ∈ ℝ 𝑀 = (𝑒𝐹𝑣))
99 fveq2 6892 . . . . . . . 8 (𝑧 = βŸ¨π‘’, π‘£βŸ© β†’ (πΉβ€˜π‘§) = (πΉβ€˜βŸ¨π‘’, π‘£βŸ©))
100 df-ov 7412 . . . . . . . 8 (𝑒𝐹𝑣) = (πΉβ€˜βŸ¨π‘’, π‘£βŸ©)
10199, 100eqtr4di 2791 . . . . . . 7 (𝑧 = βŸ¨π‘’, π‘£βŸ© β†’ (πΉβ€˜π‘§) = (𝑒𝐹𝑣))
102101eqeq2d 2744 . . . . . 6 (𝑧 = βŸ¨π‘’, π‘£βŸ© β†’ (𝑀 = (πΉβ€˜π‘§) ↔ 𝑀 = (𝑒𝐹𝑣)))
103102rexxp 5843 . . . . 5 (βˆƒπ‘§ ∈ (ℝ Γ— ℝ)𝑀 = (πΉβ€˜π‘§) ↔ βˆƒπ‘’ ∈ ℝ βˆƒπ‘£ ∈ ℝ 𝑀 = (𝑒𝐹𝑣))
10498, 103sylibr 233 . . . 4 (𝑀 ∈ 𝑅 β†’ βˆƒπ‘§ ∈ (ℝ Γ— ℝ)𝑀 = (πΉβ€˜π‘§))
105104rgen 3064 . . 3 βˆ€π‘€ ∈ 𝑅 βˆƒπ‘§ ∈ (ℝ Γ— ℝ)𝑀 = (πΉβ€˜π‘§)
106 dffo3 7104 . . 3 (𝐹:(ℝ Γ— ℝ)–onto→𝑅 ↔ (𝐹:(ℝ Γ— ℝ)βŸΆπ‘… ∧ βˆ€π‘€ ∈ 𝑅 βˆƒπ‘§ ∈ (ℝ Γ— ℝ)𝑀 = (πΉβ€˜π‘§)))
10725, 105, 106mpbir2an 710 . 2 𝐹:(ℝ Γ— ℝ)–onto→𝑅
108 df-f1o 6551 . 2 (𝐹:(ℝ Γ— ℝ)–1-1-onto→𝑅 ↔ (𝐹:(ℝ Γ— ℝ)–1-1→𝑅 ∧ 𝐹:(ℝ Γ— ℝ)–onto→𝑅))
10972, 107, 108mpbir2an 710 1 𝐹:(ℝ Γ— ℝ)–1-1-onto→𝑅
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071  {cpr 4631  βŸ¨cop 4635   Γ— cxp 5675   Fn wfn 6539  βŸΆwf 6540  β€“1-1β†’wf1 6541  β€“ontoβ†’wfo 6542  β€“1-1-ontoβ†’wf1o 6543  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  1st c1st 7973  2nd c2nd 7974   ↑m cmap 8820  β„cr 11109  1c1 11111  2c2 12267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-po 5589  df-so 5590  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-2 12275
This theorem is referenced by:  rrx2xpreen  47405  rrx2plordisom  47409
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