|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > r1funlim | Structured version Visualization version GIF version | ||
| Description: The cumulative hierarchy of sets function is a function on a limit ordinal. (This weak form of r1fnon 9808 avoids ax-rep 5278.) (Contributed by Mario Carneiro, 16-Nov-2014.) | 
| Ref | Expression | 
|---|---|
| r1funlim | ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rdgfun 8457 | . . 3 ⊢ Fun rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) | |
| 2 | df-r1 9805 | . . . 4 ⊢ 𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) | |
| 3 | 2 | funeqi 6586 | . . 3 ⊢ (Fun 𝑅1 ↔ Fun rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)) | 
| 4 | 1, 3 | mpbir 231 | . 2 ⊢ Fun 𝑅1 | 
| 5 | rdgdmlim 8458 | . . 3 ⊢ Lim dom rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) | |
| 6 | 2 | dmeqi 5914 | . . . 4 ⊢ dom 𝑅1 = dom rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) | 
| 7 | limeq 6395 | . . . 4 ⊢ (dom 𝑅1 = dom rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) → (Lim dom 𝑅1 ↔ Lim dom rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅))) | |
| 8 | 6, 7 | ax-mp 5 | . . 3 ⊢ (Lim dom 𝑅1 ↔ Lim dom rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)) | 
| 9 | 5, 8 | mpbir 231 | . 2 ⊢ Lim dom 𝑅1 | 
| 10 | 4, 9 | pm3.2i 470 | 1 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 Vcvv 3479 ∅c0 4332 𝒫 cpw 4599 ↦ cmpt 5224 dom cdm 5684 Lim wlim 6384 Fun wfun 6554 reccrdg 8450 𝑅1cr1 9803 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-r1 9805 | 
| This theorem is referenced by: r1limg 9812 r1fin 9814 r1tr 9817 r1ordg 9819 r1ord3g 9820 r1pwss 9825 r1val1 9827 rankwflemb 9834 r1elwf 9837 rankr1ai 9839 rankdmr1 9842 rankr1ag 9843 rankr1bg 9844 r1elssi 9846 pwwf 9848 unwf 9851 rankr1clem 9861 rankr1c 9862 rankval3b 9867 rankonidlem 9869 onssr1 9872 rankeq0b 9901 ackbij2 10283 wunom 10761 | 
| Copyright terms: Public domain | W3C validator |