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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 8047 | . . . 4 ⊢ Lim dom rec(𝐹, 𝐴) | |
2 | limomss 7579 | . . . 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 7595 | . . 3 ⊢ ∅ ∈ ω | |
5 | 3, 4 | sselii 3964 | . 2 ⊢ ∅ ∈ dom rec(𝐹, 𝐴) |
6 | eqid 2821 | . . 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 8049 | . . 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 8036 | . 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 1533 ∈ wcel 2110 Vcvv 3495 ⊆ wss 3936 ∅c0 4291 ifcif 4467 ∪ cuni 4832 ↦ cmpt 5139 dom cdm 5550 ran crn 5551 Lim wlim 6187 ‘cfv 6350 ωcom 7574 reccrdg 8039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 |
This theorem is referenced by: rdg0g 8057 seqomlem1 8080 seqomlem3 8082 om0 8136 oe0 8141 oev2 8142 r10 9191 aleph0 9486 ackbij2lem2 9656 ackbij2lem3 9657 satfv0 32600 satf00 32616 rdgprc 33034 finxp0 34666 finxp1o 34667 finxpreclem4 34669 finxpreclem6 34671 |
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