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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 7401 | . 2 ⊢ rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) Fn On | |
2 | df-r1 8510 | . . 3 ⊢ 𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) | |
3 | 2 | fneq1i 5899 | . 2 ⊢ (𝑅1 Fn On ↔ rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) Fn On) |
4 | 1, 3 | mpbir 220 | 1 ⊢ 𝑅1 Fn On |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3173 ∅c0 3874 𝒫 cpw 4108 ↦ cmpt 4643 Oncon0 5640 Fn wfn 5799 reccrdg 7392 𝑅1cr1 8508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-r1 8510 |
This theorem is referenced by: r1suc 8516 r1lim 8518 r111 8521 r1ord 8526 r1ord3 8528 r1elss 8552 jech9.3 8560 onwf 8576 ssrankr1 8581 r1val3 8584 r1pw 8591 rankuni 8609 rankr1b 8610 r1om 8949 hsmexlem6 9136 smobeth 9287 wunr1om 9420 r1limwun 9437 r1wunlim 9438 tskr1om 9468 tskr1om2 9469 inar1 9476 rankcf 9478 inatsk 9479 r1tskina 9483 grur1 9521 grothomex 9530 aomclem4 36645 |
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