![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mbfmptcl | Structured version Visualization version GIF version |
Description: Lemma for the MblFn predicate applied to a mapping operation. (Contributed by Mario Carneiro, 11-Aug-2014.) |
Ref | Expression |
---|---|
mbfmptcl.1 | β’ (π β (π₯ β π΄ β¦ π΅) β MblFn) |
mbfmptcl.2 | β’ ((π β§ π₯ β π΄) β π΅ β π) |
Ref | Expression |
---|---|
mbfmptcl | β’ ((π β§ π₯ β π΄) β π΅ β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mbfmptcl.1 | . . . 4 β’ (π β (π₯ β π΄ β¦ π΅) β MblFn) | |
2 | mbff 25133 | . . . 4 β’ ((π₯ β π΄ β¦ π΅) β MblFn β (π₯ β π΄ β¦ π΅):dom (π₯ β π΄ β¦ π΅)βΆβ) | |
3 | 1, 2 | syl 17 | . . 3 β’ (π β (π₯ β π΄ β¦ π΅):dom (π₯ β π΄ β¦ π΅)βΆβ) |
4 | mbfmptcl.2 | . . . . . 6 β’ ((π β§ π₯ β π΄) β π΅ β π) | |
5 | 4 | ralrimiva 3146 | . . . . 5 β’ (π β βπ₯ β π΄ π΅ β π) |
6 | dmmptg 6238 | . . . . 5 β’ (βπ₯ β π΄ π΅ β π β dom (π₯ β π΄ β¦ π΅) = π΄) | |
7 | 5, 6 | syl 17 | . . . 4 β’ (π β dom (π₯ β π΄ β¦ π΅) = π΄) |
8 | 7 | feq2d 6700 | . . 3 β’ (π β ((π₯ β π΄ β¦ π΅):dom (π₯ β π΄ β¦ π΅)βΆβ β (π₯ β π΄ β¦ π΅):π΄βΆβ)) |
9 | 3, 8 | mpbid 231 | . 2 β’ (π β (π₯ β π΄ β¦ π΅):π΄βΆβ) |
10 | 9 | fvmptelcdm 7109 | 1 β’ ((π β§ π₯ β π΄) β π΅ β β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βwral 3061 β¦ cmpt 5230 dom cdm 5675 βΆwf 6536 βcc 11104 MblFncmbf 25122 |
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 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 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-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 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-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 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-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-pm 8819 df-mbf 25127 |
This theorem is referenced by: mbfss 25154 mbfneg 25158 mbfmulc2 25171 mbflim 25176 itgcnlem 25298 itgcnval 25308 itgre 25309 itgim 25310 iblneg 25311 itgneg 25312 iblss 25313 iblss2 25314 ibladd 25329 iblsub 25330 itgadd 25333 itgsub 25334 itgfsum 25335 iblabs 25337 iblabsr 25338 iblmulc2 25339 itgmulc2 25342 itgabs 25343 itgsplit 25344 bddmulibl 25347 itgcn 25353 ditgswap 25367 ditgsplitlem 25368 ftc1a 25545 ibladdnc 36533 itgaddnc 36536 iblsubnc 36537 itgsubnc 36538 iblabsnc 36540 iblmulc2nc 36541 itgmulc2nc 36544 itgabsnc 36545 |
Copyright terms: Public domain | W3C validator |