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Mirrors > Home > ILE Home > Th. List > rdgexggg | GIF version |
Description: The recursive definition generator produces a set on a set input. (Contributed by Jim Kingdon, 4-Jul-2019.) |
Ref | Expression |
---|---|
rdgexggg | ⊢ ((𝐹 Fn V ∧ A ∈ 𝑉 ∧ B ∈ 𝑊) → (rec(𝐹, A)‘B) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-irdg 5897 | . . 3 ⊢ rec(𝐹, A) = recs((g ∈ V ↦ (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x))))) | |
2 | rdgruledefgg 5902 | . . . 4 ⊢ ((𝐹 Fn V ∧ A ∈ 𝑉) → (Fun (g ∈ V ↦ (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x)))) ∧ ((g ∈ V ↦ (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x))))‘y) ∈ V)) | |
3 | 2 | alrimiv 1751 | . . 3 ⊢ ((𝐹 Fn V ∧ A ∈ 𝑉) → ∀y(Fun (g ∈ V ↦ (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x)))) ∧ ((g ∈ V ↦ (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x))))‘y) ∈ V)) |
4 | 1, 3 | tfrex 5895 | . 2 ⊢ (((𝐹 Fn V ∧ A ∈ 𝑉) ∧ B ∈ 𝑊) → (rec(𝐹, A)‘B) ∈ V) |
5 | 4 | 3impa 1098 | 1 ⊢ ((𝐹 Fn V ∧ A ∈ 𝑉 ∧ B ∈ 𝑊) → (rec(𝐹, A)‘B) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∧ w3a 884 ∈ wcel 1390 Vcvv 2551 ∪ cun 2909 ∪ ciun 3648 ↦ cmpt 3809 dom cdm 4288 Fun wfun 4839 Fn wfn 4840 ‘cfv 4845 reccrdg 5896 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-id 4021 df-iord 4069 df-on 4071 df-suc 4074 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-recs 5861 df-irdg 5897 |
This theorem is referenced by: rdgexgg 5905 rdgisucinc 5912 omv 5974 oeiv 5975 |
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