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Mirrors > Home > MPE Home > Th. List > Mathboxes > msubval | Structured version Visualization version GIF version |
Description: A substitution applied to an expression. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
msubffval.v | β’ π = (mVRβπ) |
msubffval.r | β’ π = (mRExβπ) |
msubffval.s | β’ π = (mSubstβπ) |
msubffval.e | β’ πΈ = (mExβπ) |
msubffval.o | β’ π = (mRSubstβπ) |
Ref | Expression |
---|---|
msubval | β’ ((πΉ:π΄βΆπ β§ π΄ β π β§ π β πΈ) β ((πβπΉ)βπ) = β¨(1st βπ), ((πβπΉ)β(2nd βπ))β©) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | msubffval.v | . . . 4 β’ π = (mVRβπ) | |
2 | msubffval.r | . . . 4 β’ π = (mRExβπ) | |
3 | msubffval.s | . . . 4 β’ π = (mSubstβπ) | |
4 | msubffval.e | . . . 4 β’ πΈ = (mExβπ) | |
5 | msubffval.o | . . . 4 β’ π = (mRSubstβπ) | |
6 | 1, 2, 3, 4, 5 | msubfval 35042 | . . 3 β’ ((πΉ:π΄βΆπ β§ π΄ β π) β (πβπΉ) = (π β πΈ β¦ β¨(1st βπ), ((πβπΉ)β(2nd βπ))β©)) |
7 | 6 | 3adant3 1129 | . 2 β’ ((πΉ:π΄βΆπ β§ π΄ β π β§ π β πΈ) β (πβπΉ) = (π β πΈ β¦ β¨(1st βπ), ((πβπΉ)β(2nd βπ))β©)) |
8 | simpr 484 | . . . 4 β’ (((πΉ:π΄βΆπ β§ π΄ β π β§ π β πΈ) β§ π = π) β π = π) | |
9 | 8 | fveq2d 6888 | . . 3 β’ (((πΉ:π΄βΆπ β§ π΄ β π β§ π β πΈ) β§ π = π) β (1st βπ) = (1st βπ)) |
10 | 8 | fveq2d 6888 | . . . 4 β’ (((πΉ:π΄βΆπ β§ π΄ β π β§ π β πΈ) β§ π = π) β (2nd βπ) = (2nd βπ)) |
11 | 10 | fveq2d 6888 | . . 3 β’ (((πΉ:π΄βΆπ β§ π΄ β π β§ π β πΈ) β§ π = π) β ((πβπΉ)β(2nd βπ)) = ((πβπΉ)β(2nd βπ))) |
12 | 9, 11 | opeq12d 4876 | . 2 β’ (((πΉ:π΄βΆπ β§ π΄ β π β§ π β πΈ) β§ π = π) β β¨(1st βπ), ((πβπΉ)β(2nd βπ))β© = β¨(1st βπ), ((πβπΉ)β(2nd βπ))β©) |
13 | simp3 1135 | . 2 β’ ((πΉ:π΄βΆπ β§ π΄ β π β§ π β πΈ) β π β πΈ) | |
14 | opex 5457 | . . 3 β’ β¨(1st βπ), ((πβπΉ)β(2nd βπ))β© β V | |
15 | 14 | a1i 11 | . 2 β’ ((πΉ:π΄βΆπ β§ π΄ β π β§ π β πΈ) β β¨(1st βπ), ((πβπΉ)β(2nd βπ))β© β V) |
16 | 7, 12, 13, 15 | fvmptd 6998 | 1 β’ ((πΉ:π΄βΆπ β§ π΄ β π β§ π β πΈ) β ((πβπΉ)βπ) = β¨(1st βπ), ((πβπΉ)β(2nd βπ))β©) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 Vcvv 3468 β wss 3943 β¨cop 4629 β¦ cmpt 5224 βΆwf 6532 βcfv 6536 1st c1st 7969 2nd c2nd 7970 mVRcmvar 34979 mRExcmrex 34984 mExcmex 34985 mRSubstcmrsub 34988 mSubstcmsub 34989 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-pm 8822 df-msub 35009 |
This theorem is referenced by: msubrsub 35044 msubty 35045 msubff1 35074 |
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