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Theorem rrx2xpref1o 47491
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 8058 . . . 4 𝐹 Fn (ℝ Γ— ℝ)
4 1st2nd2 8016 . . . . . . . . 9 (𝑧 ∈ (ℝ Γ— ℝ) β†’ 𝑧 = ⟨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
54fveq2d 6894 . . . . . . . 8 (𝑧 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘§) = (πΉβ€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩))
6 df-ov 7414 . . . . . . . 8 ((1st β€˜π‘§)𝐹(2nd β€˜π‘§)) = (πΉβ€˜βŸ¨(1st β€˜π‘§), (2nd β€˜π‘§)⟩)
75, 6eqtr4di 2788 . . . . . . 7 (𝑧 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘§) = ((1st β€˜π‘§)𝐹(2nd β€˜π‘§)))
8 xp1st 8009 . . . . . . . 8 (𝑧 ∈ (ℝ Γ— ℝ) β†’ (1st β€˜π‘§) ∈ ℝ)
9 xp2nd 8010 . . . . . . . 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 7570 . . . . . . . 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 582 . . . . . . 7 (𝑧 ∈ (ℝ Γ— ℝ) β†’ ((1st β€˜π‘§)𝐹(2nd β€˜π‘§)) = {⟨1, (1st β€˜π‘§)⟩, ⟨2, (2nd β€˜π‘§)⟩})
177, 16eqtrd 2770 . . . . . 6 (𝑧 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘§) = {⟨1, (1st β€˜π‘§)⟩, ⟨2, (2nd β€˜π‘§)⟩})
18 eqid 2730 . . . . . . . 8 {1, 2} = {1, 2}
19 rrx2xpreen.r . . . . . . . 8 𝑅 = (ℝ ↑m {1, 2})
2018, 19prelrrx2 47486 . . . . . . 7 (((1st β€˜π‘§) ∈ ℝ ∧ (2nd β€˜π‘§) ∈ ℝ) β†’ {⟨1, (1st β€˜π‘§)⟩, ⟨2, (2nd β€˜π‘§)⟩} ∈ 𝑅)
218, 9, 20syl2anc 582 . . . . . 6 (𝑧 ∈ (ℝ Γ— ℝ) β†’ {⟨1, (1st β€˜π‘§)⟩, ⟨2, (2nd β€˜π‘§)⟩} ∈ 𝑅)
2217, 21eqeltrd 2831 . . . . 5 (𝑧 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘§) ∈ 𝑅)
2322rgen 3061 . . . 4 βˆ€π‘§ ∈ (ℝ Γ— ℝ)(πΉβ€˜π‘§) ∈ 𝑅
24 ffnfv 7119 . . . 4 (𝐹:(ℝ Γ— ℝ)βŸΆπ‘… ↔ (𝐹 Fn (ℝ Γ— ℝ) ∧ βˆ€π‘§ ∈ (ℝ Γ— ℝ)(πΉβ€˜π‘§) ∈ 𝑅))
253, 23, 24mpbir2an 707 . . 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 11214 . . . . . . . . . . . 12 1 ∈ V
32 fvex 6903 . . . . . . . . . . . 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 495 . . . . . . . . . 10 (⟨1, (1st β€˜π‘§)⟩ = ⟨1, (1st β€˜π‘€)⟩ β†’ (1st β€˜π‘§) = (1st β€˜π‘€))
35 2ex 12293 . . . . . . . . . . . 12 2 ∈ V
36 fvex 6903 . . . . . . . . . . . 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 495 . . . . . . . . . 10 (⟨2, (2nd β€˜π‘§)⟩ = ⟨2, (2nd β€˜π‘€)⟩ β†’ (2nd β€˜π‘§) = (2nd β€˜π‘€))
3934, 38anim12i 611 . . . . . . . . 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 12424 . . . . . . . . . . 11 1 β‰  2
44 eqneqall 2949 . . . . . . . . . . 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 722 . . . . . . . . 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 596 . . . . . . . 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 853 . . . . . . 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 8016 . . . . . . . . 9 (𝑀 ∈ (ℝ Γ— ℝ) β†’ 𝑀 = ⟨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)
5251fveq2d 6894 . . . . . . . 8 (𝑀 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘€) = (πΉβ€˜βŸ¨(1st β€˜π‘€), (2nd β€˜π‘€)⟩))
53 df-ov 7414 . . . . . . . 8 ((1st β€˜π‘€)𝐹(2nd β€˜π‘€)) = (πΉβ€˜βŸ¨(1st β€˜π‘€), (2nd β€˜π‘€)⟩)
5452, 53eqtr4di 2788 . . . . . . 7 (𝑀 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘€) = ((1st β€˜π‘€)𝐹(2nd β€˜π‘€)))
55 xp1st 8009 . . . . . . . 8 (𝑀 ∈ (ℝ Γ— ℝ) β†’ (1st β€˜π‘€) ∈ ℝ)
56 xp2nd 8010 . . . . . . . 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 7570 . . . . . . . 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 582 . . . . . . 7 (𝑀 ∈ (ℝ Γ— ℝ) β†’ ((1st β€˜π‘€)𝐹(2nd β€˜π‘€)) = {⟨1, (1st β€˜π‘€)⟩, ⟨2, (2nd β€˜π‘€)⟩})
6454, 63eqtrd 2770 . . . . . 6 (𝑀 ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘€) = {⟨1, (1st β€˜π‘€)⟩, ⟨2, (2nd β€˜π‘€)⟩})
6517, 64eqeqan12d 2744 . . . . 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 2744 . . . . . 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 3195 . . 3 βˆ€π‘§ ∈ (ℝ Γ— ℝ)βˆ€π‘€ ∈ (ℝ Γ— ℝ)((πΉβ€˜π‘§) = (πΉβ€˜π‘€) β†’ 𝑧 = 𝑀)
71 dff13 7256 . . 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 707 . 2 𝐹:(ℝ Γ— ℝ)–1-1→𝑅
7319eleq2i 2823 . . . . . . . 8 (𝑀 ∈ 𝑅 ↔ 𝑀 ∈ (ℝ ↑m {1, 2}))
74 reex 11203 . . . . . . . . 9 ℝ ∈ V
75 prex 5431 . . . . . . . . 9 {1, 2} ∈ V
7674, 75elmap 8867 . . . . . . . 8 (𝑀 ∈ (ℝ ↑m {1, 2}) ↔ 𝑀:{1, 2}βŸΆβ„)
77 1re 11218 . . . . . . . . 9 1 ∈ ℝ
78 2re 12290 . . . . . . . . 9 2 ∈ ℝ
79 fpr2g 7214 . . . . . . . . 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 688 . . . . . . . 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 2741 . . . . . . . 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 2741 . . . . . . . 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 7570 . . . . . . . 8 ((𝑒 ∈ ℝ ∧ 𝑣 ∈ ℝ) β†’ (𝑒𝐹𝑣) = {⟨1, π‘’βŸ©, ⟨2, π‘£βŸ©})
9695eqeq2d 2741 . . . . . . 7 ((𝑒 ∈ ℝ ∧ 𝑣 ∈ ℝ) β†’ (𝑀 = (𝑒𝐹𝑣) ↔ 𝑀 = {⟨1, π‘’βŸ©, ⟨2, π‘£βŸ©}))
97962rexbiia 3213 . . . . . 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 6890 . . . . . . . 8 (𝑧 = βŸ¨π‘’, π‘£βŸ© β†’ (πΉβ€˜π‘§) = (πΉβ€˜βŸ¨π‘’, π‘£βŸ©))
100 df-ov 7414 . . . . . . . 8 (𝑒𝐹𝑣) = (πΉβ€˜βŸ¨π‘’, π‘£βŸ©)
10199, 100eqtr4di 2788 . . . . . . 7 (𝑧 = βŸ¨π‘’, π‘£βŸ© β†’ (πΉβ€˜π‘§) = (𝑒𝐹𝑣))
102101eqeq2d 2741 . . . . . 6 (𝑧 = βŸ¨π‘’, π‘£βŸ© β†’ (𝑀 = (πΉβ€˜π‘§) ↔ 𝑀 = (𝑒𝐹𝑣)))
103102rexxp 5841 . . . . 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 3061 . . 3 βˆ€π‘€ ∈ 𝑅 βˆƒπ‘§ ∈ (ℝ Γ— ℝ)𝑀 = (πΉβ€˜π‘§)
106 dffo3 7102 . . 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 707 . 2 𝐹:(ℝ Γ— ℝ)–onto→𝑅
108 df-f1o 6549 . 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 707 1 𝐹:(ℝ Γ— ℝ)–1-1-onto→𝑅
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  βˆƒwrex 3068  {cpr 4629  βŸ¨cop 4633   Γ— cxp 5673   Fn wfn 6537  βŸΆwf 6538  β€“1-1β†’wf1 6539  β€“ontoβ†’wfo 6540  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7411   ∈ cmpo 7413  1st c1st 7975  2nd c2nd 7976   ↑m cmap 8822  β„cr 11111  1c1 11113  2c2 12271
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  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 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-2 12279
This theorem is referenced by:  rrx2xpreen  47492  rrx2plordisom  47496
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