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Theorem rrx2xpref1o 46504
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 5388 . . . . 5 {⟨1, π‘₯⟩, ⟨2, π‘¦βŸ©} ∈ V
31, 2fnmpoi 7991 . . . 4 𝐹 Fn (ℝ Γ— ℝ)
4 1st2nd2 7951 . . . . . . . . 9 (𝑧 ∈ (ℝ Γ— ℝ) β†’ 𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
54fveq2d 6842 . . . . . . . 8 (𝑧 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘§) = (πΉβ€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩))
6 df-ov 7353 . . . . . . . 8 ((1st β€˜π‘§)𝐹(2nd β€˜π‘§)) = (πΉβ€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
75, 6eqtr4di 2796 . . . . . . 7 (𝑧 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘§) = ((1st β€˜π‘§)𝐹(2nd β€˜π‘§)))
8 xp1st 7944 . . . . . . . 8 (𝑧 ∈ (ℝ Γ— ℝ) β†’ (1st β€˜π‘§) ∈ ℝ)
9 xp2nd 7945 . . . . . . . 8 (𝑧 ∈ (ℝ Γ— ℝ) β†’ (2nd β€˜π‘§) ∈ ℝ)
10 opeq2 4830 . . . . . . . . . 10 (π‘₯ = (1st β€˜π‘§) β†’ ⟨1, π‘₯⟩ = ⟨1, (1st β€˜π‘§)⟩)
1110preq1d 4699 . . . . . . . . 9 (π‘₯ = (1st β€˜π‘§) β†’ {⟨1, π‘₯⟩, ⟨2, π‘¦βŸ©} = {⟨1, (1st β€˜π‘§)⟩, ⟨2, π‘¦βŸ©})
12 opeq2 4830 . . . . . . . . . 10 (𝑦 = (2nd β€˜π‘§) β†’ ⟨2, π‘¦βŸ© = ⟨2, (2nd β€˜π‘§)⟩)
1312preq2d 4700 . . . . . . . . 9 (𝑦 = (2nd β€˜π‘§) β†’ {⟨1, (1st β€˜π‘§)⟩, ⟨2, π‘¦βŸ©} = {⟨1, (1st β€˜π‘§)⟩, ⟨2, (2nd β€˜π‘§)⟩})
14 prex 5388 . . . . . . . . 9 {⟨1, (1st β€˜π‘§)⟩, ⟨2, (2nd β€˜π‘§)⟩} ∈ V
1511, 13, 1, 14ovmpo 7508 . . . . . . . 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 2778 . . . . . 6 (𝑧 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘§) = {⟨1, (1st β€˜π‘§)⟩, ⟨2, (2nd β€˜π‘§)⟩})
18 eqid 2738 . . . . . . . 8 {1, 2} = {1, 2}
19 rrx2xpreen.r . . . . . . . 8 𝑅 = (ℝ ↑m {1, 2})
2018, 19prelrrx2 46499 . . . . . . 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 2839 . . . . 5 (𝑧 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘§) ∈ 𝑅)
2322rgen 3065 . . . 4 βˆ€π‘§ ∈ (ℝ Γ— ℝ)(πΉβ€˜π‘§) ∈ 𝑅
24 ffnfv 7061 . . . 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 5420 . . . . . . . 8 ⟨1, (1st β€˜π‘§)⟩ ∈ V
27 opex 5420 . . . . . . . 8 ⟨2, (2nd β€˜π‘§)⟩ ∈ V
28 opex 5420 . . . . . . . 8 ⟨1, (1st β€˜π‘€)⟩ ∈ V
29 opex 5420 . . . . . . . 8 ⟨2, (2nd β€˜π‘€)⟩ ∈ V
3026, 27, 28, 29preq12b 4807 . . . . . . 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 11085 . . . . . . . . . . . 12 1 ∈ V
32 fvex 6851 . . . . . . . . . . . 12 (1st β€˜π‘§) ∈ V
3331, 32opth 5432 . . . . . . . . . . 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 12164 . . . . . . . . . . . 12 2 ∈ V
36 fvex 6851 . . . . . . . . . . . 12 (2nd β€˜π‘§) ∈ V
3735, 36opth 5432 . . . . . . . . . . 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 5432 . . . . . . . . 9 (⟨1, (1st β€˜π‘§)⟩ = ⟨2, (2nd β€˜π‘€)⟩ ↔ (1 = 2 ∧ (1st β€˜π‘§) = (2nd β€˜π‘€)))
4235, 36opth 5432 . . . . . . . . 9 (⟨2, (2nd β€˜π‘§)⟩ = ⟨1, (1st β€˜π‘€)⟩ ↔ (2 = 1 ∧ (2nd β€˜π‘§) = (1st β€˜π‘€)))
43 1ne2 12295 . . . . . . . . . . 11 1 β‰  2
44 eqneqall 2953 . . . . . . . . . . 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 7951 . . . . . . . . 9 (𝑀 ∈ (ℝ Γ— ℝ) β†’ 𝑀 = ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)
5251fveq2d 6842 . . . . . . . 8 (𝑀 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘€) = (πΉβ€˜βŸ¨(1st β€˜π‘€), (2nd β€˜π‘€)⟩))
53 df-ov 7353 . . . . . . . 8 ((1st β€˜π‘€)𝐹(2nd β€˜π‘€)) = (πΉβ€˜βŸ¨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)
5452, 53eqtr4di 2796 . . . . . . 7 (𝑀 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘€) = ((1st β€˜π‘€)𝐹(2nd β€˜π‘€)))
55 xp1st 7944 . . . . . . . 8 (𝑀 ∈ (ℝ Γ— ℝ) β†’ (1st β€˜π‘€) ∈ ℝ)
56 xp2nd 7945 . . . . . . . 8 (𝑀 ∈ (ℝ Γ— ℝ) β†’ (2nd β€˜π‘€) ∈ ℝ)
57 opeq2 4830 . . . . . . . . . 10 (π‘₯ = (1st β€˜π‘€) β†’ ⟨1, π‘₯⟩ = ⟨1, (1st β€˜π‘€)⟩)
5857preq1d 4699 . . . . . . . . 9 (π‘₯ = (1st β€˜π‘€) β†’ {⟨1, π‘₯⟩, ⟨2, π‘¦βŸ©} = {⟨1, (1st β€˜π‘€)⟩, ⟨2, π‘¦βŸ©})
59 opeq2 4830 . . . . . . . . . 10 (𝑦 = (2nd β€˜π‘€) β†’ ⟨2, π‘¦βŸ© = ⟨2, (2nd β€˜π‘€)⟩)
6059preq2d 4700 . . . . . . . . 9 (𝑦 = (2nd β€˜π‘€) β†’ {⟨1, (1st β€˜π‘€)⟩, ⟨2, π‘¦βŸ©} = {⟨1, (1st β€˜π‘€)⟩, ⟨2, (2nd β€˜π‘€)⟩})
61 prex 5388 . . . . . . . . 9 {⟨1, (1st β€˜π‘€)⟩, ⟨2, (2nd β€˜π‘€)⟩} ∈ V
6258, 60, 1, 61ovmpo 7508 . . . . . . . 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 2778 . . . . . 6 (𝑀 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘€) = {⟨1, (1st β€˜π‘€)⟩, ⟨2, (2nd β€˜π‘€)⟩})
6517, 64eqeqan12d 2752 . . . . 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 2752 . . . . . 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 5432 . . . . . 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 3193 . . 3 βˆ€π‘§ ∈ (ℝ Γ— ℝ)βˆ€π‘€ ∈ (ℝ Γ— ℝ)((πΉβ€˜π‘§) = (πΉβ€˜π‘€) β†’ 𝑧 = 𝑀)
71 dff13 7197 . . 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 2830 . . . . . . . 8 (𝑀 ∈ 𝑅 ↔ 𝑀 ∈ (ℝ ↑m {1, 2}))
74 reex 11076 . . . . . . . . 9 ℝ ∈ V
75 prex 5388 . . . . . . . . 9 {1, 2} ∈ V
7674, 75elmap 8743 . . . . . . . 8 (𝑀 ∈ (ℝ ↑m {1, 2}) ↔ 𝑀:{1, 2}βŸΆβ„)
77 1re 11089 . . . . . . . . 9 1 ∈ ℝ
78 2re 12161 . . . . . . . . 9 2 ∈ ℝ
79 fpr2g 7156 . . . . . . . . 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 4830 . . . . . . . . . 10 (𝑒 = (π‘€β€˜1) β†’ ⟨1, π‘’βŸ© = ⟨1, (π‘€β€˜1)⟩)
8382preq1d 4699 . . . . . . . . 9 (𝑒 = (π‘€β€˜1) β†’ {⟨1, π‘’βŸ©, ⟨2, π‘£βŸ©} = {⟨1, (π‘€β€˜1)⟩, ⟨2, π‘£βŸ©})
8483eqeq2d 2749 . . . . . . . 8 (𝑒 = (π‘€β€˜1) β†’ (𝑀 = {⟨1, π‘’βŸ©, ⟨2, π‘£βŸ©} ↔ 𝑀 = {⟨1, (π‘€β€˜1)⟩, ⟨2, π‘£βŸ©}))
85 opeq2 4830 . . . . . . . . . 10 (𝑣 = (π‘€β€˜2) β†’ ⟨2, π‘£βŸ© = ⟨2, (π‘€β€˜2)⟩)
8685preq2d 4700 . . . . . . . . 9 (𝑣 = (π‘€β€˜2) β†’ {⟨1, (π‘€β€˜1)⟩, ⟨2, π‘£βŸ©} = {⟨1, (π‘€β€˜1)⟩, ⟨2, (π‘€β€˜2)⟩})
8786eqeq2d 2749 . . . . . . . 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 3591 . . . . . . 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 4830 . . . . . . . . . 10 (π‘₯ = 𝑒 β†’ ⟨1, π‘₯⟩ = ⟨1, π‘’βŸ©)
9190preq1d 4699 . . . . . . . . 9 (π‘₯ = 𝑒 β†’ {⟨1, π‘₯⟩, ⟨2, π‘¦βŸ©} = {⟨1, π‘’βŸ©, ⟨2, π‘¦βŸ©})
92 opeq2 4830 . . . . . . . . . 10 (𝑦 = 𝑣 β†’ ⟨2, π‘¦βŸ© = ⟨2, π‘£βŸ©)
9392preq2d 4700 . . . . . . . . 9 (𝑦 = 𝑣 β†’ {⟨1, π‘’βŸ©, ⟨2, π‘¦βŸ©} = {⟨1, π‘’βŸ©, ⟨2, π‘£βŸ©})
94 prex 5388 . . . . . . . . 9 {⟨1, π‘’βŸ©, ⟨2, π‘£βŸ©} ∈ V
9591, 93, 1, 94ovmpo 7508 . . . . . . . 8 ((𝑒 ∈ ℝ ∧ 𝑣 ∈ ℝ) β†’ (𝑒𝐹𝑣) = {⟨1, π‘’βŸ©, ⟨2, π‘£βŸ©})
9695eqeq2d 2749 . . . . . . 7 ((𝑒 ∈ ℝ ∧ 𝑣 ∈ ℝ) β†’ (𝑀 = (𝑒𝐹𝑣) ↔ 𝑀 = {⟨1, π‘’βŸ©, ⟨2, π‘£βŸ©}))
97962rexbiia 3208 . . . . . 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 6838 . . . . . . . 8 (𝑧 = βŸ¨π‘’, π‘£βŸ© β†’ (πΉβ€˜π‘§) = (πΉβ€˜βŸ¨π‘’, π‘£βŸ©))
100 df-ov 7353 . . . . . . . 8 (𝑒𝐹𝑣) = (πΉβ€˜βŸ¨π‘’, π‘£βŸ©)
10199, 100eqtr4di 2796 . . . . . . 7 (𝑧 = βŸ¨π‘’, π‘£βŸ© β†’ (πΉβ€˜π‘§) = (𝑒𝐹𝑣))
102101eqeq2d 2749 . . . . . 6 (𝑧 = βŸ¨π‘’, π‘£βŸ© β†’ (𝑀 = (πΉβ€˜π‘§) ↔ 𝑀 = (𝑒𝐹𝑣)))
103102rexxp 5795 . . . . 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 3065 . . 3 βˆ€π‘€ ∈ 𝑅 βˆƒπ‘§ ∈ (ℝ Γ— ℝ)𝑀 = (πΉβ€˜π‘§)
106 dffo3 7047 . . 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 6499 . 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 2942  βˆ€wral 3063  βˆƒwrex 3072  {cpr 4587  βŸ¨cop 4591   Γ— cxp 5629   Fn wfn 6487  βŸΆwf 6488  β€“1-1β†’wf1 6489  β€“ontoβ†’wfo 6490  β€“1-1-ontoβ†’wf1o 6491  β€˜cfv 6492  (class class class)co 7350   ∈ cmpo 7352  1st c1st 7910  2nd c2nd 7911   ↑m cmap 8699  β„cr 10984  1c1 10986  2c2 12142
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 2709  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7663  ax-cnex 11041  ax-resscn 11042  ax-1cn 11043  ax-icn 11044  ax-addcl 11045  ax-addrcl 11046  ax-mulcl 11047  ax-mulrcl 11048  ax-mulcom 11049  ax-addass 11050  ax-mulass 11051  ax-distr 11052  ax-i2m1 11053  ax-1ne0 11054  ax-1rid 11055  ax-rnegex 11056  ax-rrecex 11057  ax-cnre 11058  ax-pre-lttri 11059  ax-pre-lttrn 11060  ax-pre-ltadd 11061  ax-pre-mulgt0 11062
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 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-po 5543  df-so 5544  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7306  df-ov 7353  df-oprab 7354  df-mpo 7355  df-1st 7912  df-2nd 7913  df-er 8582  df-map 8701  df-en 8818  df-dom 8819  df-sdom 8820  df-pnf 11125  df-mnf 11126  df-xr 11127  df-ltxr 11128  df-le 11129  df-sub 11321  df-neg 11322  df-2 12150
This theorem is referenced by:  rrx2xpreen  46505  rrx2plordisom  46509
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