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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 25142 | . . . 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 3147 | . . . . 5 β’ (π β βπ₯ β π΄ π΅ β π) |
6 | dmmptg 6242 | . . . . 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 6704 | . . 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 7113 | 1 β’ ((π β§ π₯ β π΄) β π΅ β β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3062 β¦ cmpt 5232 dom cdm 5677 βΆwf 6540 βcc 11108 MblFncmbf 25131 |
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-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 |
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 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-pm 8823 df-mbf 25136 |
This theorem is referenced by: mbfss 25163 mbfneg 25167 mbfmulc2 25180 mbflim 25185 itgcnlem 25307 itgcnval 25317 itgre 25318 itgim 25319 iblneg 25320 itgneg 25321 iblss 25322 iblss2 25323 ibladd 25338 iblsub 25339 itgadd 25342 itgsub 25343 itgfsum 25344 iblabs 25346 iblabsr 25347 iblmulc2 25348 itgmulc2 25351 itgabs 25352 itgsplit 25353 bddmulibl 25356 itgcn 25362 ditgswap 25376 ditgsplitlem 25377 ftc1a 25554 ibladdnc 36545 itgaddnc 36548 iblsubnc 36549 itgsubnc 36550 iblabsnc 36552 iblmulc2nc 36553 itgmulc2nc 36556 itgabsnc 36557 |
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