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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 25149 | . . . 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 6241 | . . . . 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 6703 | . . 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 7114 | 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 5231 dom cdm 5676 βΆwf 6539 βcc 11110 MblFncmbf 25138 |
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 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 |
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 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-pm 8825 df-mbf 25143 |
This theorem is referenced by: mbfss 25170 mbfneg 25174 mbfmulc2 25187 mbflim 25192 itgcnlem 25314 itgcnval 25324 itgre 25325 itgim 25326 iblneg 25327 itgneg 25328 iblss 25329 iblss2 25330 ibladd 25345 iblsub 25346 itgadd 25349 itgsub 25350 itgfsum 25351 iblabs 25353 iblabsr 25354 iblmulc2 25355 itgmulc2 25358 itgabs 25359 itgsplit 25360 bddmulibl 25363 itgcn 25369 ditgswap 25383 ditgsplitlem 25384 ftc1a 25561 ibladdnc 36631 itgaddnc 36634 iblsubnc 36635 itgsubnc 36636 iblabsnc 36638 iblmulc2nc 36639 itgmulc2nc 36642 itgabsnc 36643 |
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