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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 9762 avoids ax-rep 5286.) (Contributed by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
r1funlim | ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdgfun 8416 | . . 3 ⊢ Fun rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) | |
2 | df-r1 9759 | . . . 4 ⊢ 𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) | |
3 | 2 | funeqi 6570 | . . 3 ⊢ (Fun 𝑅1 ↔ Fun rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)) |
4 | 1, 3 | mpbir 230 | . 2 ⊢ Fun 𝑅1 |
5 | rdgdmlim 8417 | . . 3 ⊢ Lim dom rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) | |
6 | 2 | dmeqi 5905 | . . . 4 ⊢ dom 𝑅1 = dom rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) |
7 | limeq 6377 | . . . 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 230 | . 2 ⊢ Lim dom 𝑅1 |
10 | 4, 9 | pm3.2i 472 | 1 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 Vcvv 3475 ∅c0 4323 𝒫 cpw 4603 ↦ cmpt 5232 dom cdm 5677 Lim wlim 6366 Fun wfun 6538 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-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-lim 6370 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: r1limg 9766 r1fin 9768 r1tr 9771 r1ordg 9773 r1ord3g 9774 r1pwss 9779 r1val1 9781 rankwflemb 9788 r1elwf 9791 rankr1ai 9793 rankdmr1 9796 rankr1ag 9797 rankr1bg 9798 r1elssi 9800 pwwf 9802 unwf 9805 rankr1clem 9815 rankr1c 9816 rankval3b 9821 rankonidlem 9823 onssr1 9826 rankeq0b 9855 ackbij2 10238 wunom 10715 |
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