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Mirrors > Home > MPE Home > Th. List > r1fnon | Structured version Visualization version GIF version |
Description: The cumulative hierarchy of sets function is a function on the class of ordinal numbers. (Contributed by NM, 5-Oct-2003.) (Revised by Mario Carneiro, 10-Sep-2013.) |
Ref | Expression |
---|---|
r1fnon | ⊢ 𝑅1 Fn On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdgfnon 8418 | . 2 ⊢ rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) Fn On | |
2 | df-r1 9759 | . . 3 ⊢ 𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) | |
3 | 2 | fneq1i 6647 | . 2 ⊢ (𝑅1 Fn On ↔ rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) Fn On) |
4 | 1, 3 | mpbir 230 | 1 ⊢ 𝑅1 Fn On |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3475 ∅c0 4323 𝒫 cpw 4603 ↦ cmpt 5232 Oncon0 6365 Fn wfn 6539 reccrdg 8409 𝑅1cr1 9757 |
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-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 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-pred 6301 df-ord 6368 df-on 6369 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-r1 9759 |
This theorem is referenced by: r1suc 9765 r1lim 9767 r111 9770 r1ord 9775 r1ord3 9777 r1elss 9801 jech9.3 9809 onwf 9825 ssrankr1 9830 r1val3 9833 r1pw 9840 rankuni 9858 rankr1b 9859 r1om 10239 hsmexlem6 10426 smobeth 10581 wunr1om 10714 r1limwun 10731 r1wunlim 10732 tskr1om 10762 tskr1om2 10763 inar1 10770 rankcf 10772 inatsk 10773 r1tskina 10777 grur1 10815 grothomex 10824 aomclem4 41799 grur1cld 42991 |
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