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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 9759 avoids ax-rep 5285.) (Contributed by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
r1funlim | ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdgfun 8413 | . . 3 ⊢ Fun rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) | |
2 | df-r1 9756 | . . . 4 ⊢ 𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) | |
3 | 2 | funeqi 6567 | . . 3 ⊢ (Fun 𝑅1 ↔ Fun rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)) |
4 | 1, 3 | mpbir 230 | . 2 ⊢ Fun 𝑅1 |
5 | rdgdmlim 8414 | . . 3 ⊢ Lim dom rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) | |
6 | 2 | dmeqi 5903 | . . . 4 ⊢ dom 𝑅1 = dom rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) |
7 | limeq 6374 | . . . 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 4322 𝒫 cpw 4602 ↦ cmpt 5231 dom cdm 5676 Lim wlim 6363 Fun wfun 6535 reccrdg 8406 𝑅1cr1 9754 |
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 5299 ax-nul 5306 ax-pr 5427 ax-un 7722 |
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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7409 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-r1 9756 |
This theorem is referenced by: r1limg 9763 r1fin 9765 r1tr 9768 r1ordg 9770 r1ord3g 9771 r1pwss 9776 r1val1 9778 rankwflemb 9785 r1elwf 9788 rankr1ai 9790 rankdmr1 9793 rankr1ag 9794 rankr1bg 9795 r1elssi 9797 pwwf 9799 unwf 9802 rankr1clem 9812 rankr1c 9813 rankval3b 9818 rankonidlem 9820 onssr1 9823 rankeq0b 9852 ackbij2 10235 wunom 10712 |
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