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Mirrors > Home > ILE Home > Th. List > rdgival | GIF version |
Description: Value of the recursive definition generator. (Contributed by Jim Kingdon, 26-Jul-2019.) |
Ref | Expression |
---|---|
rdgival | ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdgivallem 6146 | . 2 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)))) | |
2 | fvres 5329 | . . . . 5 ⊢ (𝑥 ∈ 𝐵 → ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) = (rec(𝐹, 𝐴)‘𝑥)) | |
3 | 2 | fveq2d 5309 | . . . 4 ⊢ (𝑥 ∈ 𝐵 → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) = (𝐹‘(rec(𝐹, 𝐴)‘𝑥))) |
4 | 3 | iuneq2i 3748 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) = ∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)) |
5 | 4 | uneq2i 3151 | . 2 ⊢ (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥))) |
6 | 1, 5 | syl6eq 2136 | 1 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘(rec(𝐹, 𝐴)‘𝑥)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 924 = wceq 1289 ∈ wcel 1438 Vcvv 2619 ∪ cun 2997 ∪ ciun 3730 Oncon0 4190 ↾ cres 4440 Fn wfn 5010 ‘cfv 5015 reccrdg 6134 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-coll 3954 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-iun 3732 df-br 3846 df-opab 3900 df-mpt 3901 df-tr 3937 df-id 4120 df-iord 4193 df-on 4195 df-suc 4198 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-fv 5023 df-recs 6070 df-irdg 6135 |
This theorem is referenced by: rdgss 6148 rdgisuc1 6149 rdgisucinc 6150 oav2 6224 omv2 6226 |
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