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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 8513 avoids ax-rep 4699.) (Contributed by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
r1funlim | ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdgfun 7399 | . . 3 ⊢ Fun rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) | |
2 | df-r1 8510 | . . . 4 ⊢ 𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) | |
3 | 2 | funeqi 5824 | . . 3 ⊢ (Fun 𝑅1 ↔ Fun rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)) |
4 | 1, 3 | mpbir 220 | . 2 ⊢ Fun 𝑅1 |
5 | rdgdmlim 7400 | . . 3 ⊢ Lim dom rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) | |
6 | 2 | dmeqi 5247 | . . . 4 ⊢ dom 𝑅1 = dom rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) |
7 | limeq 5652 | . . . 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 220 | . 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 195 ∧ wa 383 = wceq 1475 Vcvv 3173 ∅c0 3874 𝒫 cpw 4108 ↦ cmpt 4643 dom cdm 5038 Lim wlim 5641 Fun wfun 5798 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-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-pw 4110 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-lim 5645 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: r1limg 8517 r1fin 8519 r1tr 8522 r1ordg 8524 r1ord3g 8525 r1pwss 8530 r1val1 8532 rankwflemb 8539 r1elwf 8542 rankr1ai 8544 rankdmr1 8547 rankr1ag 8548 rankr1bg 8549 r1elssi 8551 pwwf 8553 unwf 8556 rankr1clem 8566 rankr1c 8567 rankval3b 8572 rankonidlem 8574 onssr1 8577 rankeq0b 8606 ackbij2 8948 wunom 9421 |
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