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Mirrors > Home > ILE Home > Th. List > tfrex | GIF version |
Description: The transfinite recursion function is set-like if the input is. (Contributed by Mario Carneiro, 3-Jul-2019.) |
Ref | Expression |
---|---|
tfrex.1 | ⊢ 𝐹 = recs(𝐺) |
tfrex.2 | ⊢ (𝜑 → ∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V)) |
Ref | Expression |
---|---|
tfrex | ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝐹‘𝐴) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrex.1 | . . 3 ⊢ 𝐹 = recs(𝐺) | |
2 | 1 | fveq1i 5321 | . 2 ⊢ (𝐹‘𝐴) = (recs(𝐺)‘𝐴) |
3 | eqid 2089 | . . . 4 ⊢ {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))} | |
4 | 3 | tfrlem3 6092 | . . 3 ⊢ {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} |
5 | tfrex.2 | . . 3 ⊢ (𝜑 → ∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V)) | |
6 | 4, 5 | tfrexlem 6115 | . 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (recs(𝐺)‘𝐴) ∈ V) |
7 | 2, 6 | syl5eqel 2175 | 1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝐹‘𝐴) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1288 = wceq 1290 ∈ wcel 1439 {cab 2075 ∀wral 2360 ∃wrex 2361 Vcvv 2622 Oncon0 4201 ↾ cres 4456 Fun wfun 5024 Fn wfn 5025 ‘cfv 5030 recscrecs 6085 |
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 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3962 ax-sep 3965 ax-pow 4017 ax-pr 4047 ax-un 4271 ax-setind 4368 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2624 df-sbc 2844 df-csb 2937 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-nul 3290 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-iun 3740 df-br 3854 df-opab 3908 df-mpt 3909 df-tr 3945 df-id 4131 df-iord 4204 df-on 4206 df-suc 4209 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-rn 4465 df-res 4466 df-ima 4467 df-iota 4995 df-fun 5032 df-fn 5033 df-f 5034 df-f1 5035 df-fo 5036 df-f1o 5037 df-fv 5038 df-recs 6086 |
This theorem is referenced by: rdgexggg 6158 |
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