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Mirrors > Home > MPE Home > Th. List > r1funlim | Structured version Visualization version Unicode version |
Description: The cumulative hierarchy of sets function is a function on a limit ordinal. (This weak form of r1fnon 8263 avoids ax-rep 4528.) (Contributed by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
r1funlim |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdgfun 7159 |
. . 3
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2 | df-r1 8260 |
. . . 4
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3 | 2 | funeqi 5620 |
. . 3
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4 | 1, 3 | mpbir 214 |
. 2
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5 | rdgdmlim 7160 |
. . 3
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6 | 2 | dmeqi 5054 |
. . . 4
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7 | limeq 5453 |
. . . 4
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8 | 6, 7 | ax-mp 5 |
. . 3
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9 | 5, 8 | mpbir 214 |
. 2
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10 | 4, 9 | pm3.2i 461 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-8 1899 ax-9 1906 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 ax-sep 4538 ax-nul 4547 ax-pow 4594 ax-pr 4652 ax-un 6609 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1457 df-ex 1674 df-nf 1678 df-sb 1808 df-eu 2313 df-mo 2314 df-clab 2448 df-cleq 2454 df-clel 2457 df-nfc 2591 df-ne 2634 df-ral 2753 df-rex 2754 df-reu 2755 df-rab 2757 df-v 3058 df-sbc 3279 df-csb 3375 df-dif 3418 df-un 3420 df-in 3422 df-ss 3429 df-pss 3431 df-nul 3743 df-if 3893 df-pw 3964 df-sn 3980 df-pr 3982 df-tp 3984 df-op 3986 df-uni 4212 df-iun 4293 df-br 4416 df-opab 4475 df-mpt 4476 df-tr 4511 df-eprel 4763 df-id 4767 df-po 4773 df-so 4774 df-fr 4811 df-we 4813 df-xp 4858 df-rel 4859 df-cnv 4860 df-co 4861 df-dm 4862 df-rn 4863 df-res 4864 df-ima 4865 df-pred 5398 df-ord 5444 df-on 5445 df-lim 5446 df-suc 5447 df-iota 5564 df-fun 5602 df-fn 5603 df-f 5604 df-f1 5605 df-fo 5606 df-f1o 5607 df-fv 5608 df-wrecs 7053 df-recs 7115 df-rdg 7153 df-r1 8260 |
This theorem is referenced by: r1limg 8267 r1fin 8269 r1tr 8272 r1ordg 8274 r1ord3g 8275 r1pwss 8280 r1val1 8282 rankwflemb 8289 r1elwf 8292 rankr1ai 8294 rankdmr1 8297 rankr1ag 8298 rankr1bg 8299 r1elssi 8301 pwwf 8303 unwf 8306 rankr1clem 8316 rankr1c 8317 rankval3b 8322 rankonidlem 8324 onssr1 8327 rankeq0b 8356 ackbij2 8698 wunom 9170 |
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