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Mirrors > Home > MPE Home > Th. List > ucnimalem | Structured version Visualization version GIF version |
Description: Reformulate the πΊ function as a mapping with one variable. (Contributed by Thierry Arnoux, 19-Nov-2017.) |
Ref | Expression |
---|---|
ucnprima.1 | β’ (π β π β (UnifOnβπ)) |
ucnprima.2 | β’ (π β π β (UnifOnβπ)) |
ucnprima.3 | β’ (π β πΉ β (π Cnuπ)) |
ucnprima.4 | β’ (π β π β π) |
ucnprima.5 | β’ πΊ = (π₯ β π, π¦ β π β¦ β¨(πΉβπ₯), (πΉβπ¦)β©) |
Ref | Expression |
---|---|
ucnimalem | β’ πΊ = (π β (π Γ π) β¦ β¨(πΉβ(1st βπ)), (πΉβ(2nd βπ))β©) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ucnprima.5 | . 2 β’ πΊ = (π₯ β π, π¦ β π β¦ β¨(πΉβπ₯), (πΉβπ¦)β©) | |
2 | vex 3467 | . . . . . 6 β’ π₯ β V | |
3 | vex 3467 | . . . . . 6 β’ π¦ β V | |
4 | 2, 3 | op1std 8001 | . . . . 5 β’ (π = β¨π₯, π¦β© β (1st βπ) = π₯) |
5 | 4 | fveq2d 6896 | . . . 4 β’ (π = β¨π₯, π¦β© β (πΉβ(1st βπ)) = (πΉβπ₯)) |
6 | 2, 3 | op2ndd 8002 | . . . . 5 β’ (π = β¨π₯, π¦β© β (2nd βπ) = π¦) |
7 | 6 | fveq2d 6896 | . . . 4 β’ (π = β¨π₯, π¦β© β (πΉβ(2nd βπ)) = (πΉβπ¦)) |
8 | 5, 7 | opeq12d 4877 | . . 3 β’ (π = β¨π₯, π¦β© β β¨(πΉβ(1st βπ)), (πΉβ(2nd βπ))β© = β¨(πΉβπ₯), (πΉβπ¦)β©) |
9 | 8 | mpompt 7531 | . 2 β’ (π β (π Γ π) β¦ β¨(πΉβ(1st βπ)), (πΉβ(2nd βπ))β©) = (π₯ β π, π¦ β π β¦ β¨(πΉβπ₯), (πΉβπ¦)β©) |
10 | 1, 9 | eqtr4i 2756 | 1 β’ πΊ = (π β (π Γ π) β¦ β¨(πΉβ(1st βπ)), (πΉβ(2nd βπ))β©) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β¨cop 4630 β¦ cmpt 5226 Γ cxp 5670 βcfv 6543 (class class class)co 7416 β cmpo 7418 1st c1st 7989 2nd c2nd 7990 UnifOncust 24122 Cnucucn 24198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6495 df-fun 6545 df-fv 6551 df-oprab 7420 df-mpo 7421 df-1st 7991 df-2nd 7992 |
This theorem is referenced by: ucnima 24204 |
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