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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 9180 avoids ax-rep 5154.) (Contributed by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
r1funlim | ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdgfun 8035 | . . 3 ⊢ Fun rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) | |
2 | df-r1 9177 | . . . 4 ⊢ 𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) | |
3 | 2 | funeqi 6345 | . . 3 ⊢ (Fun 𝑅1 ↔ Fun rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)) |
4 | 1, 3 | mpbir 234 | . 2 ⊢ Fun 𝑅1 |
5 | rdgdmlim 8036 | . . 3 ⊢ Lim dom rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) | |
6 | 2 | dmeqi 5737 | . . . 4 ⊢ dom 𝑅1 = dom rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) |
7 | limeq 6171 | . . . 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 234 | . 2 ⊢ Lim dom 𝑅1 |
10 | 4, 9 | pm3.2i 474 | 1 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 Vcvv 3441 ∅c0 4243 𝒫 cpw 4497 ↦ cmpt 5110 dom cdm 5519 Lim wlim 6160 Fun wfun 6318 reccrdg 8028 𝑅1cr1 9175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-r1 9177 |
This theorem is referenced by: r1limg 9184 r1fin 9186 r1tr 9189 r1ordg 9191 r1ord3g 9192 r1pwss 9197 r1val1 9199 rankwflemb 9206 r1elwf 9209 rankr1ai 9211 rankdmr1 9214 rankr1ag 9215 rankr1bg 9216 r1elssi 9218 pwwf 9220 unwf 9223 rankr1clem 9233 rankr1c 9234 rankval3b 9239 rankonidlem 9241 onssr1 9244 rankeq0b 9273 ackbij2 9654 wunom 10131 |
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