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Theorem rrx2xpref1o 47357
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 5431 . . . . 5 {⟨1, π‘₯⟩, ⟨2, π‘¦βŸ©} ∈ V
31, 2fnmpoi 8052 . . . 4 𝐹 Fn (ℝ Γ— ℝ)
4 1st2nd2 8010 . . . . . . . . 9 (𝑧 ∈ (ℝ Γ— ℝ) β†’ 𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
54fveq2d 6892 . . . . . . . 8 (𝑧 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘§) = (πΉβ€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩))
6 df-ov 7408 . . . . . . . 8 ((1st β€˜π‘§)𝐹(2nd β€˜π‘§)) = (πΉβ€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
75, 6eqtr4di 2790 . . . . . . 7 (𝑧 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘§) = ((1st β€˜π‘§)𝐹(2nd β€˜π‘§)))
8 xp1st 8003 . . . . . . . 8 (𝑧 ∈ (ℝ Γ— ℝ) β†’ (1st β€˜π‘§) ∈ ℝ)
9 xp2nd 8004 . . . . . . . 8 (𝑧 ∈ (ℝ Γ— ℝ) β†’ (2nd β€˜π‘§) ∈ ℝ)
10 opeq2 4873 . . . . . . . . . 10 (π‘₯ = (1st β€˜π‘§) β†’ ⟨1, π‘₯⟩ = ⟨1, (1st β€˜π‘§)⟩)
1110preq1d 4742 . . . . . . . . 9 (π‘₯ = (1st β€˜π‘§) β†’ {⟨1, π‘₯⟩, ⟨2, π‘¦βŸ©} = {⟨1, (1st β€˜π‘§)⟩, ⟨2, π‘¦βŸ©})
12 opeq2 4873 . . . . . . . . . 10 (𝑦 = (2nd β€˜π‘§) β†’ ⟨2, π‘¦βŸ© = ⟨2, (2nd β€˜π‘§)⟩)
1312preq2d 4743 . . . . . . . . 9 (𝑦 = (2nd β€˜π‘§) β†’ {⟨1, (1st β€˜π‘§)⟩, ⟨2, π‘¦βŸ©} = {⟨1, (1st β€˜π‘§)⟩, ⟨2, (2nd β€˜π‘§)⟩})
14 prex 5431 . . . . . . . . 9 {⟨1, (1st β€˜π‘§)⟩, ⟨2, (2nd β€˜π‘§)⟩} ∈ V
1511, 13, 1, 14ovmpo 7564 . . . . . . . 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 584 . . . . . . 7 (𝑧 ∈ (ℝ Γ— ℝ) β†’ ((1st β€˜π‘§)𝐹(2nd β€˜π‘§)) = {⟨1, (1st β€˜π‘§)⟩, ⟨2, (2nd β€˜π‘§)⟩})
177, 16eqtrd 2772 . . . . . 6 (𝑧 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘§) = {⟨1, (1st β€˜π‘§)⟩, ⟨2, (2nd β€˜π‘§)⟩})
18 eqid 2732 . . . . . . . 8 {1, 2} = {1, 2}
19 rrx2xpreen.r . . . . . . . 8 𝑅 = (ℝ ↑m {1, 2})
2018, 19prelrrx2 47352 . . . . . . 7 (((1st β€˜π‘§) ∈ ℝ ∧ (2nd β€˜π‘§) ∈ ℝ) β†’ {⟨1, (1st β€˜π‘§)⟩, ⟨2, (2nd β€˜π‘§)⟩} ∈ 𝑅)
218, 9, 20syl2anc 584 . . . . . 6 (𝑧 ∈ (ℝ Γ— ℝ) β†’ {⟨1, (1st β€˜π‘§)⟩, ⟨2, (2nd β€˜π‘§)⟩} ∈ 𝑅)
2217, 21eqeltrd 2833 . . . . 5 (𝑧 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘§) ∈ 𝑅)
2322rgen 3063 . . . 4 βˆ€π‘§ ∈ (ℝ Γ— ℝ)(πΉβ€˜π‘§) ∈ 𝑅
24 ffnfv 7114 . . . 4 (𝐹:(ℝ Γ— ℝ)βŸΆπ‘… ↔ (𝐹 Fn (ℝ Γ— ℝ) ∧ βˆ€π‘§ ∈ (ℝ Γ— ℝ)(πΉβ€˜π‘§) ∈ 𝑅))
253, 23, 24mpbir2an 709 . . 3 𝐹:(ℝ Γ— ℝ)βŸΆπ‘…
26 opex 5463 . . . . . . . 8 ⟨1, (1st β€˜π‘§)⟩ ∈ V
27 opex 5463 . . . . . . . 8 ⟨2, (2nd β€˜π‘§)⟩ ∈ V
28 opex 5463 . . . . . . . 8 ⟨1, (1st β€˜π‘€)⟩ ∈ V
29 opex 5463 . . . . . . . 8 ⟨2, (2nd β€˜π‘€)⟩ ∈ V
3026, 27, 28, 29preq12b 4850 . . . . . . 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 11206 . . . . . . . . . . . 12 1 ∈ V
32 fvex 6901 . . . . . . . . . . . 12 (1st β€˜π‘§) ∈ V
3331, 32opth 5475 . . . . . . . . . . 11 (⟨1, (1st β€˜π‘§)⟩ = ⟨1, (1st β€˜π‘€)⟩ ↔ (1 = 1 ∧ (1st β€˜π‘§) = (1st β€˜π‘€)))
3433simprbi 497 . . . . . . . . . 10 (⟨1, (1st β€˜π‘§)⟩ = ⟨1, (1st β€˜π‘€)⟩ β†’ (1st β€˜π‘§) = (1st β€˜π‘€))
35 2ex 12285 . . . . . . . . . . . 12 2 ∈ V
36 fvex 6901 . . . . . . . . . . . 12 (2nd β€˜π‘§) ∈ V
3735, 36opth 5475 . . . . . . . . . . 11 (⟨2, (2nd β€˜π‘§)⟩ = ⟨2, (2nd β€˜π‘€)⟩ ↔ (2 = 2 ∧ (2nd β€˜π‘§) = (2nd β€˜π‘€)))
3837simprbi 497 . . . . . . . . . 10 (⟨2, (2nd β€˜π‘§)⟩ = ⟨2, (2nd β€˜π‘€)⟩ β†’ (2nd β€˜π‘§) = (2nd β€˜π‘€))
3934, 38anim12i 613 . . . . . . . . 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 5475 . . . . . . . . 9 (⟨1, (1st β€˜π‘§)⟩ = ⟨2, (2nd β€˜π‘€)⟩ ↔ (1 = 2 ∧ (1st β€˜π‘§) = (2nd β€˜π‘€)))
4235, 36opth 5475 . . . . . . . . 9 (⟨2, (2nd β€˜π‘§)⟩ = ⟨1, (1st β€˜π‘€)⟩ ↔ (2 = 1 ∧ (2nd β€˜π‘§) = (1st β€˜π‘€)))
43 1ne2 12416 . . . . . . . . . . 11 1 β‰  2
44 eqneqall 2951 . . . . . . . . . . 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 724 . . . . . . . . 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 598 . . . . . . . 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 855 . . . . . . 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 8010 . . . . . . . . 9 (𝑀 ∈ (ℝ Γ— ℝ) β†’ 𝑀 = ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)
5251fveq2d 6892 . . . . . . . 8 (𝑀 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘€) = (πΉβ€˜βŸ¨(1st β€˜π‘€), (2nd β€˜π‘€)⟩))
53 df-ov 7408 . . . . . . . 8 ((1st β€˜π‘€)𝐹(2nd β€˜π‘€)) = (πΉβ€˜βŸ¨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)
5452, 53eqtr4di 2790 . . . . . . 7 (𝑀 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘€) = ((1st β€˜π‘€)𝐹(2nd β€˜π‘€)))
55 xp1st 8003 . . . . . . . 8 (𝑀 ∈ (ℝ Γ— ℝ) β†’ (1st β€˜π‘€) ∈ ℝ)
56 xp2nd 8004 . . . . . . . 8 (𝑀 ∈ (ℝ Γ— ℝ) β†’ (2nd β€˜π‘€) ∈ ℝ)
57 opeq2 4873 . . . . . . . . . 10 (π‘₯ = (1st β€˜π‘€) β†’ ⟨1, π‘₯⟩ = ⟨1, (1st β€˜π‘€)⟩)
5857preq1d 4742 . . . . . . . . 9 (π‘₯ = (1st β€˜π‘€) β†’ {⟨1, π‘₯⟩, ⟨2, π‘¦βŸ©} = {⟨1, (1st β€˜π‘€)⟩, ⟨2, π‘¦βŸ©})
59 opeq2 4873 . . . . . . . . . 10 (𝑦 = (2nd β€˜π‘€) β†’ ⟨2, π‘¦βŸ© = ⟨2, (2nd β€˜π‘€)⟩)
6059preq2d 4743 . . . . . . . . 9 (𝑦 = (2nd β€˜π‘€) β†’ {⟨1, (1st β€˜π‘€)⟩, ⟨2, π‘¦βŸ©} = {⟨1, (1st β€˜π‘€)⟩, ⟨2, (2nd β€˜π‘€)⟩})
61 prex 5431 . . . . . . . . 9 {⟨1, (1st β€˜π‘€)⟩, ⟨2, (2nd β€˜π‘€)⟩} ∈ V
6258, 60, 1, 61ovmpo 7564 . . . . . . . 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 584 . . . . . . 7 (𝑀 ∈ (ℝ Γ— ℝ) β†’ ((1st β€˜π‘€)𝐹(2nd β€˜π‘€)) = {⟨1, (1st β€˜π‘€)⟩, ⟨2, (2nd β€˜π‘€)⟩})
6454, 63eqtrd 2772 . . . . . 6 (𝑀 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘€) = {⟨1, (1st β€˜π‘€)⟩, ⟨2, (2nd β€˜π‘€)⟩})
6517, 64eqeqan12d 2746 . . . . 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 2746 . . . . . 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 5475 . . . . . 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 286 . . . . 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 293 . . . 4 ((𝑧 ∈ (ℝ Γ— ℝ) ∧ 𝑀 ∈ (ℝ Γ— ℝ)) β†’ ((πΉβ€˜π‘§) = (πΉβ€˜π‘€) β†’ 𝑧 = 𝑀))
7069rgen2 3197 . . 3 βˆ€π‘§ ∈ (ℝ Γ— ℝ)βˆ€π‘€ ∈ (ℝ Γ— ℝ)((πΉβ€˜π‘§) = (πΉβ€˜π‘€) β†’ 𝑧 = 𝑀)
71 dff13 7250 . . 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 709 . 2 𝐹:(ℝ Γ— ℝ)–1-1→𝑅
7319eleq2i 2825 . . . . . . . 8 (𝑀 ∈ 𝑅 ↔ 𝑀 ∈ (ℝ ↑m {1, 2}))
74 reex 11197 . . . . . . . . 9 ℝ ∈ V
75 prex 5431 . . . . . . . . 9 {1, 2} ∈ V
7674, 75elmap 8861 . . . . . . . 8 (𝑀 ∈ (ℝ ↑m {1, 2}) ↔ 𝑀:{1, 2}βŸΆβ„)
77 1re 11210 . . . . . . . . 9 1 ∈ ℝ
78 2re 12282 . . . . . . . . 9 2 ∈ ℝ
79 fpr2g 7209 . . . . . . . . 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 690 . . . . . . . 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 296 . . . . . . 7 (𝑀 ∈ 𝑅 ↔ ((π‘€β€˜1) ∈ ℝ ∧ (π‘€β€˜2) ∈ ℝ ∧ 𝑀 = {⟨1, (π‘€β€˜1)⟩, ⟨2, (π‘€β€˜2)⟩}))
82 opeq2 4873 . . . . . . . . . 10 (𝑒 = (π‘€β€˜1) β†’ ⟨1, π‘’βŸ© = ⟨1, (π‘€β€˜1)⟩)
8382preq1d 4742 . . . . . . . . 9 (𝑒 = (π‘€β€˜1) β†’ {⟨1, π‘’βŸ©, ⟨2, π‘£βŸ©} = {⟨1, (π‘€β€˜1)⟩, ⟨2, π‘£βŸ©})
8483eqeq2d 2743 . . . . . . . 8 (𝑒 = (π‘€β€˜1) β†’ (𝑀 = {⟨1, π‘’βŸ©, ⟨2, π‘£βŸ©} ↔ 𝑀 = {⟨1, (π‘€β€˜1)⟩, ⟨2, π‘£βŸ©}))
85 opeq2 4873 . . . . . . . . . 10 (𝑣 = (π‘€β€˜2) β†’ ⟨2, π‘£βŸ© = ⟨2, (π‘€β€˜2)⟩)
8685preq2d 4743 . . . . . . . . 9 (𝑣 = (π‘€β€˜2) β†’ {⟨1, (π‘€β€˜1)⟩, ⟨2, π‘£βŸ©} = {⟨1, (π‘€β€˜1)⟩, ⟨2, (π‘€β€˜2)⟩})
8786eqeq2d 2743 . . . . . . . 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 3623 . . . . . . 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 4873 . . . . . . . . . 10 (π‘₯ = 𝑒 β†’ ⟨1, π‘₯⟩ = ⟨1, π‘’βŸ©)
9190preq1d 4742 . . . . . . . . 9 (π‘₯ = 𝑒 β†’ {⟨1, π‘₯⟩, ⟨2, π‘¦βŸ©} = {⟨1, π‘’βŸ©, ⟨2, π‘¦βŸ©})
92 opeq2 4873 . . . . . . . . . 10 (𝑦 = 𝑣 β†’ ⟨2, π‘¦βŸ© = ⟨2, π‘£βŸ©)
9392preq2d 4743 . . . . . . . . 9 (𝑦 = 𝑣 β†’ {⟨1, π‘’βŸ©, ⟨2, π‘¦βŸ©} = {⟨1, π‘’βŸ©, ⟨2, π‘£βŸ©})
94 prex 5431 . . . . . . . . 9 {⟨1, π‘’βŸ©, ⟨2, π‘£βŸ©} ∈ V
9591, 93, 1, 94ovmpo 7564 . . . . . . . 8 ((𝑒 ∈ ℝ ∧ 𝑣 ∈ ℝ) β†’ (𝑒𝐹𝑣) = {⟨1, π‘’βŸ©, ⟨2, π‘£βŸ©})
9695eqeq2d 2743 . . . . . . 7 ((𝑒 ∈ ℝ ∧ 𝑣 ∈ ℝ) β†’ (𝑀 = (𝑒𝐹𝑣) ↔ 𝑀 = {⟨1, π‘’βŸ©, ⟨2, π‘£βŸ©}))
97962rexbiia 3215 . . . . . 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 6888 . . . . . . . 8 (𝑧 = βŸ¨π‘’, π‘£βŸ© β†’ (πΉβ€˜π‘§) = (πΉβ€˜βŸ¨π‘’, π‘£βŸ©))
100 df-ov 7408 . . . . . . . 8 (𝑒𝐹𝑣) = (πΉβ€˜βŸ¨π‘’, π‘£βŸ©)
10199, 100eqtr4di 2790 . . . . . . 7 (𝑧 = βŸ¨π‘’, π‘£βŸ© β†’ (πΉβ€˜π‘§) = (𝑒𝐹𝑣))
102101eqeq2d 2743 . . . . . 6 (𝑧 = βŸ¨π‘’, π‘£βŸ© β†’ (𝑀 = (πΉβ€˜π‘§) ↔ 𝑀 = (𝑒𝐹𝑣)))
103102rexxp 5840 . . . . 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 3063 . . 3 βˆ€π‘€ ∈ 𝑅 βˆƒπ‘§ ∈ (ℝ Γ— ℝ)𝑀 = (πΉβ€˜π‘§)
106 dffo3 7100 . . 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 709 . 2 𝐹:(ℝ Γ— ℝ)–onto→𝑅
108 df-f1o 6547 . 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 709 1 𝐹:(ℝ Γ— ℝ)–1-1-onto→𝑅
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  {cpr 4629  βŸ¨cop 4633   Γ— cxp 5673   Fn wfn 6535  βŸΆwf 6536  β€“1-1β†’wf1 6537  β€“ontoβ†’wfo 6538  β€“1-1-ontoβ†’wf1o 6539  β€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407  1st c1st 7969  2nd c2nd 7970   ↑m cmap 8816  β„cr 11105  1c1 11107  2c2 12263
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 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-po 5587  df-so 5588  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-2 12271
This theorem is referenced by:  rrx2xpreen  47358  rrx2plordisom  47362
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