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Mirrors > Home > MPE Home > Th. List > rdg0g | Structured version Visualization version GIF version |
Description: The initial value of the recursive definition generator. (Contributed by NM, 25-Apr-1995.) |
Ref | Expression |
---|---|
rdg0g | ⊢ (𝐴 ∈ 𝐶 → (rec(𝐹, 𝐴)‘∅) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdgeq2 8230 | . . . 4 ⊢ (𝑥 = 𝐴 → rec(𝐹, 𝑥) = rec(𝐹, 𝐴)) | |
2 | 1 | fveq1d 6768 | . . 3 ⊢ (𝑥 = 𝐴 → (rec(𝐹, 𝑥)‘∅) = (rec(𝐹, 𝐴)‘∅)) |
3 | id 22 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
4 | 2, 3 | eqeq12d 2754 | . 2 ⊢ (𝑥 = 𝐴 → ((rec(𝐹, 𝑥)‘∅) = 𝑥 ↔ (rec(𝐹, 𝐴)‘∅) = 𝐴)) |
5 | vex 3433 | . . 3 ⊢ 𝑥 ∈ V | |
6 | 5 | rdg0 8239 | . 2 ⊢ (rec(𝐹, 𝑥)‘∅) = 𝑥 |
7 | 4, 6 | vtoclg 3502 | 1 ⊢ (𝐴 ∈ 𝐶 → (rec(𝐹, 𝐴)‘∅) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ∅c0 4256 ‘cfv 6426 reccrdg 8227 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pr 5350 ax-un 7578 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-ov 7270 df-om 7703 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 |
This theorem is referenced by: fr0g 8254 oa0 8333 ttrclselem1 9470 ttrclselem2 9471 findreccl 34650 exrecfnlem 35558 |
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