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Mirrors > Home > ILE Home > Th. List > rdgexg | GIF version |
Description: The recursive definition generator produces a set on a set input. (Contributed by Mario Carneiro, 3-Jul-2019.) |
Ref | Expression |
---|---|
rdg0.1 | ⊢ 𝐴 ∈ V |
rdg0.2 | ⊢ 𝐹 Fn V |
Ref | Expression |
---|---|
rdgexg | ⊢ (𝐵 ∈ 𝑉 → (rec(𝐹, 𝐴)‘𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdg0.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | rdg0.2 | . . 3 ⊢ 𝐹 Fn V | |
3 | 2 | rdgexgg 6181 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (rec(𝐹, 𝐴)‘𝐵) ∈ V) |
4 | 1, 3 | mpan 416 | 1 ⊢ (𝐵 ∈ 𝑉 → (rec(𝐹, 𝐴)‘𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1445 Vcvv 2633 Fn wfn 5044 ‘cfv 5049 reccrdg 6172 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-tr 3959 df-id 4144 df-iord 4217 df-on 4219 df-suc 4222 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-recs 6108 df-irdg 6173 |
This theorem is referenced by: fnoa 6248 oaexg 6249 fnom 6251 omexg 6252 fnoei 6253 oeiexg 6254 |
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