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Mirrors > Home > MPE Home > Th. List > Mathboxes > msubty | Structured version Visualization version GIF version |
Description: The type of a substituted expression is the same as the original type. (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) |
Ref | Expression |
---|---|
msubty | β’ ((πΉ:π΄βΆπ β§ π΄ β π β§ π β πΈ) β (1st β((πβπΉ)βπ)) = (1st βπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | msubffval.v | . . 3 β’ π = (mVRβπ) | |
2 | msubffval.r | . . 3 β’ π = (mRExβπ) | |
3 | msubffval.s | . . 3 β’ π = (mSubstβπ) | |
4 | msubffval.e | . . 3 β’ πΈ = (mExβπ) | |
5 | eqid 2733 | . . 3 β’ (mRSubstβπ) = (mRSubstβπ) | |
6 | 1, 2, 3, 4, 5 | msubval 34183 | . 2 β’ ((πΉ:π΄βΆπ β§ π΄ β π β§ π β πΈ) β ((πβπΉ)βπ) = β¨(1st βπ), (((mRSubstβπ)βπΉ)β(2nd βπ))β©) |
7 | fvex 6859 | . . 3 β’ (1st βπ) β V | |
8 | fvex 6859 | . . 3 β’ (((mRSubstβπ)βπΉ)β(2nd βπ)) β V | |
9 | 7, 8 | op1std 7935 | . 2 β’ (((πβπΉ)βπ) = β¨(1st βπ), (((mRSubstβπ)βπΉ)β(2nd βπ))β© β (1st β((πβπΉ)βπ)) = (1st βπ)) |
10 | 6, 9 | syl 17 | 1 β’ ((πΉ:π΄βΆπ β§ π΄ β π β§ π β πΈ) β (1st β((πβπΉ)βπ)) = (1st βπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 β wss 3914 β¨cop 4596 βΆwf 6496 βcfv 6500 1st c1st 7923 2nd c2nd 7924 mVRcmvar 34119 mRExcmrex 34124 mExcmex 34125 mRSubstcmrsub 34128 mSubstcmsub 34129 |
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 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7925 df-pm 8774 df-msub 34149 |
This theorem is referenced by: (None) |
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