Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > rdg0 | Structured version Visualization version GIF version |
Description: The initial value of the recursive definition generator. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) |
Ref | Expression |
---|---|
rdg.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
rdg0 | ⊢ (rec(𝐹, 𝐴)‘∅) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdgdmlim 7513 | . . . 4 ⊢ Lim dom rec(𝐹, 𝐴) | |
2 | limomss 7070 | . . . 4 ⊢ (Lim dom rec(𝐹, 𝐴) → ω ⊆ dom rec(𝐹, 𝐴)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ω ⊆ dom rec(𝐹, 𝐴) |
4 | peano1 7085 | . . 3 ⊢ ∅ ∈ ω | |
5 | 3, 4 | sselii 3600 | . 2 ⊢ ∅ ∈ dom rec(𝐹, 𝐴) |
6 | eqid 2622 | . . 3 ⊢ (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))))) | |
7 | rdgvalg 7515 | . . 3 ⊢ (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))) | |
8 | rdg.1 | . . 3 ⊢ 𝐴 ∈ V | |
9 | 6, 7, 8 | tz7.44-1 7502 | . 2 ⊢ (∅ ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴)‘∅) = 𝐴) |
10 | 5, 9 | ax-mp 5 | 1 ⊢ (rec(𝐹, 𝐴)‘∅) = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 ∅c0 3915 ifcif 4086 ∪ cuni 4436 ↦ cmpt 4729 dom cdm 5114 ran crn 5115 Lim wlim 5724 ‘cfv 5888 ωcom 7065 reccrdg 7505 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 |
This theorem is referenced by: rdg0g 7523 seqomlem1 7545 seqomlem3 7547 om0 7597 oe0 7602 oev2 7603 r10 8631 aleph0 8889 ackbij2lem2 9062 ackbij2lem3 9063 rdgprc 31700 finxp0 33228 finxp1o 33229 finxpreclem4 33231 finxpreclem6 33233 |
Copyright terms: Public domain | W3C validator |