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Mirrors > Home > ILE Home > Th. List > tfr0 | GIF version |
Description: Transfinite recursion at the empty set. (Contributed by Jim Kingdon, 8-May-2020.) |
Ref | Expression |
---|---|
tfr.1 | ⊢ 𝐹 = recs(𝐺) |
Ref | Expression |
---|---|
tfr0 | ⊢ ((𝐺‘∅) ∈ 𝑉 → (𝐹‘∅) = (𝐺‘∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfr.1 | . . . 4 ⊢ 𝐹 = recs(𝐺) | |
2 | 1 | tfr0dm 6316 | . . 3 ⊢ ((𝐺‘∅) ∈ 𝑉 → ∅ ∈ dom 𝐹) |
3 | 1 | tfr2a 6315 | . . 3 ⊢ (∅ ∈ dom 𝐹 → (𝐹‘∅) = (𝐺‘(𝐹 ↾ ∅))) |
4 | 2, 3 | syl 14 | . 2 ⊢ ((𝐺‘∅) ∈ 𝑉 → (𝐹‘∅) = (𝐺‘(𝐹 ↾ ∅))) |
5 | res0 4906 | . . 3 ⊢ (𝐹 ↾ ∅) = ∅ | |
6 | 5 | fveq2i 5513 | . 2 ⊢ (𝐺‘(𝐹 ↾ ∅)) = (𝐺‘∅) |
7 | 4, 6 | eqtrdi 2226 | 1 ⊢ ((𝐺‘∅) ∈ 𝑉 → (𝐹‘∅) = (𝐺‘∅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 ∅c0 3422 dom cdm 4622 ↾ cres 4624 ‘cfv 5211 recscrecs 6298 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4289 df-iord 4362 df-on 4364 df-suc 4367 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-res 4634 df-iota 5173 df-fun 5213 df-fn 5214 df-fv 5219 df-recs 6299 |
This theorem is referenced by: rdg0 6381 frec0g 6391 |
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