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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 8036 | . . . 4 ⊢ Lim dom rec(𝐹, 𝐴) | |
2 | limomss 7565 | . . . 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 7581 | . . 3 ⊢ ∅ ∈ ω | |
5 | 3, 4 | sselii 3912 | . 2 ⊢ ∅ ∈ dom rec(𝐹, 𝐴) |
6 | eqid 2798 | . . 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 8038 | . . 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 8025 | . 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 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 ∅c0 4243 ifcif 4425 ∪ cuni 4800 ↦ cmpt 5110 dom cdm 5519 ran crn 5520 Lim wlim 6160 ‘cfv 6324 ωcom 7560 reccrdg 8028 |
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-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 |
This theorem is referenced by: rdg0g 8046 seqomlem1 8069 seqomlem3 8071 om0 8125 oe0 8130 oev2 8131 r10 9181 aleph0 9477 ackbij2lem2 9651 ackbij2lem3 9652 satfv0 32718 satf00 32734 rdgprc 33152 finxp0 34808 finxp1o 34809 finxpreclem4 34811 finxpreclem6 34813 |
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