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Mirrors > Home > MPE Home > Th. List > o1mptrcl | Structured version Visualization version GIF version |
Description: Reverse closure for an eventually bounded function. (Contributed by Mario Carneiro, 26-May-2016.) |
Ref | Expression |
---|---|
o1add2.1 | β’ ((π β§ π₯ β π΄) β π΅ β π) |
o1mptrcl.3 | β’ (π β (π₯ β π΄ β¦ π΅) β π(1)) |
Ref | Expression |
---|---|
o1mptrcl | β’ ((π β§ π₯ β π΄) β π΅ β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | o1mptrcl.3 | . . . 4 β’ (π β (π₯ β π΄ β¦ π΅) β π(1)) | |
2 | o1f 15417 | . . . 4 β’ ((π₯ β π΄ β¦ π΅) 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 3140 | . . . . 5 β’ (π β βπ₯ β π΄ π΅ β π) |
6 | dmmptg 6195 | . . . . 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 6655 | . . 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 7062 | 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 3061 β¦ cmpt 5189 dom cdm 5634 βΆwf 6493 βcc 11054 π(1)co1 15374 |
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 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 |
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-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-pm 8771 df-o1 15378 |
This theorem is referenced by: o1le 15543 fsumo1 15702 o1fsum 15703 o1cxp 26340 mulogsum 26896 |
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