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Mirrors > Home > MPE Home > Th. List > rdglimg | Structured version Visualization version GIF version |
Description: The value of the recursive definition generator at a limit ordinal. (Contributed by NM, 16-Nov-2014.) |
Ref | Expression |
---|---|
rdglimg | ⊢ ((𝐵 ∈ dom rec(𝐹, 𝐴) ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ (rec(𝐹, 𝐴) “ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . 2 ⊢ (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐹‘(𝑥‘∪ dom 𝑥))))) = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐹‘(𝑥‘∪ dom 𝑥))))) | |
2 | rdgvalg 8440 | . 2 ⊢ (𝑦 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴)‘𝑦) = ((𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐹‘(𝑥‘∪ dom 𝑥)))))‘(rec(𝐹, 𝐴) ↾ 𝑦))) | |
3 | rdgseg 8443 | . 2 ⊢ (𝑦 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴) ↾ 𝑦) ∈ V) | |
4 | rdgfun 8437 | . . 3 ⊢ Fun rec(𝐹, 𝐴) | |
5 | funfn 6584 | . . 3 ⊢ (Fun rec(𝐹, 𝐴) ↔ rec(𝐹, 𝐴) Fn dom rec(𝐹, 𝐴)) | |
6 | 4, 5 | mpbi 229 | . 2 ⊢ rec(𝐹, 𝐴) Fn dom rec(𝐹, 𝐴) |
7 | rdgdmlim 8438 | . . 3 ⊢ Lim dom rec(𝐹, 𝐴) | |
8 | limord 6431 | . . 3 ⊢ (Lim dom rec(𝐹, 𝐴) → Ord dom rec(𝐹, 𝐴)) | |
9 | 7, 8 | ax-mp 5 | . 2 ⊢ Ord dom rec(𝐹, 𝐴) |
10 | 1, 2, 3, 6, 9 | tz7.44-3 8429 | 1 ⊢ ((𝐵 ∈ dom rec(𝐹, 𝐴) ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ (rec(𝐹, 𝐴) “ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3461 ∅c0 4322 ifcif 4530 ∪ cuni 4909 ↦ cmpt 5232 dom cdm 5678 ran crn 5679 “ cima 5681 Ord word 6370 Lim wlim 6372 Fun wfun 6543 Fn wfn 6544 ‘cfv 6549 reccrdg 8430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 |
This theorem is referenced by: rdglim 8447 r1limg 9801 |
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