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Mirrors > Home > ILE Home > Th. List > tfr2a | GIF version |
Description: A weak version of transfinite recursion. (Contributed by Mario Carneiro, 24-Jun-2015.) |
Ref | Expression |
---|---|
tfr.1 | ⊢ 𝐹 = recs(𝐺) |
Ref | Expression |
---|---|
tfr2a | ⊢ (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2040 | . . . 4 ⊢ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} | |
2 | 1 | tfrlem9 5935 | . . 3 ⊢ (𝐴 ∈ dom recs(𝐺) → (recs(𝐺)‘𝐴) = (𝐺‘(recs(𝐺) ↾ 𝐴))) |
3 | tfr.1 | . . . 4 ⊢ 𝐹 = recs(𝐺) | |
4 | 3 | dmeqi 4536 | . . 3 ⊢ dom 𝐹 = dom recs(𝐺) |
5 | 2, 4 | eleq2s 2132 | . 2 ⊢ (𝐴 ∈ dom 𝐹 → (recs(𝐺)‘𝐴) = (𝐺‘(recs(𝐺) ↾ 𝐴))) |
6 | 3 | fveq1i 5179 | . 2 ⊢ (𝐹‘𝐴) = (recs(𝐺)‘𝐴) |
7 | 3 | reseq1i 4608 | . . 3 ⊢ (𝐹 ↾ 𝐴) = (recs(𝐺) ↾ 𝐴) |
8 | 7 | fveq2i 5181 | . 2 ⊢ (𝐺‘(𝐹 ↾ 𝐴)) = (𝐺‘(recs(𝐺) ↾ 𝐴)) |
9 | 5, 6, 8 | 3eqtr4g 2097 | 1 ⊢ (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = (𝐺‘(𝐹 ↾ 𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 {cab 2026 ∀wral 2306 ∃wrex 2307 Oncon0 4100 dom cdm 4345 ↾ cres 4347 Fn wfn 4897 ‘cfv 4902 recscrecs 5919 |
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-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-res 4357 df-iota 4867 df-fun 4904 df-fn 4905 df-fv 4910 df-recs 5920 |
This theorem is referenced by: tfr0 5937 tfri2d 5950 tfri2 5952 |
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