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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 8420 | . 2 ⊢ rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) Fn On | |
2 | df-r1 9761 | . . 3 ⊢ 𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) | |
3 | 2 | fneq1i 6645 | . 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 3472 ∅c0 4321 𝒫 cpw 4601 ↦ cmpt 5230 Oncon0 6363 Fn wfn 6537 reccrdg 8411 𝑅1cr1 9759 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-r1 9761 |
This theorem is referenced by: r1suc 9767 r1lim 9769 r111 9772 r1ord 9777 r1ord3 9779 r1elss 9803 jech9.3 9811 onwf 9827 ssrankr1 9832 r1val3 9835 r1pw 9842 rankuni 9860 rankr1b 9861 r1om 10241 hsmexlem6 10428 smobeth 10583 wunr1om 10716 r1limwun 10733 r1wunlim 10734 tskr1om 10764 tskr1om2 10765 inar1 10772 rankcf 10774 inatsk 10775 r1tskina 10779 grur1 10817 grothomex 10826 aomclem4 42101 grur1cld 43293 |
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