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Mirrors > Home > MPE Home > Th. List > lo1mptrcl | Structured version Visualization version GIF version |
Description: Reverse closure for an eventually upper bounded function. (Contributed by Mario Carneiro, 26-May-2016.) |
Ref | Expression |
---|---|
o1add2.1 | β’ ((π β§ π₯ β π΄) β π΅ β π) |
lo1mptrcl.3 | β’ (π β (π₯ β π΄ β¦ π΅) β β€π(1)) |
Ref | Expression |
---|---|
lo1mptrcl | β’ ((π β§ π₯ β π΄) β π΅ β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lo1mptrcl.3 | . . . 4 β’ (π β (π₯ β π΄ β¦ π΅) β β€π(1)) | |
2 | lo1f 15494 | . . . 4 β’ ((π₯ β π΄ β¦ π΅) β β€π(1) 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 | o1add2.1 | . . . . . 6 β’ ((π β§ π₯ β π΄) β π΅ β π) | |
5 | 4 | ralrimiva 3136 | . . . . 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 7118 | 1 β’ ((π β§ π₯ β π΄) β π΅ β β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 βwral 3051 β¦ cmpt 5226 dom cdm 5672 βΆwf 6539 βcr 11137 β€π(1)clo1 15463 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 df-pm 8846 df-lo1 15467 |
This theorem is referenced by: lo1add 15603 lo1mul 15604 lo1mul2 15605 lo1sub 15607 lo1le 15630 |
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