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Mirrors > Home > ILE Home > Th. List > frec2uz0d | GIF version |
Description: The mapping 𝐺 is a one-to-one mapping from ω onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number 𝐶 (normally 0 for the upper integers ℕ0 or 1 for the upper integers ℕ), 1 maps to 𝐶 + 1, etc. This theorem shows the value of 𝐺 at ordinal natural number zero. (Contributed by Jim Kingdon, 16-May-2020.) |
Ref | Expression |
---|---|
frec2uz.1 | ⊢ (𝜑 → 𝐶 ∈ ℤ) |
frec2uz.2 | ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) |
Ref | Expression |
---|---|
frec2uz0d | ⊢ (𝜑 → (𝐺‘∅) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frec2uz.2 | . . 3 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶) | |
2 | 1 | fveq1i 5253 | . 2 ⊢ (𝐺‘∅) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘∅) |
3 | frec2uz.1 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℤ) | |
4 | frec0g 6093 | . . 3 ⊢ (𝐶 ∈ ℤ → (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘∅) = 𝐶) | |
5 | 3, 4 | syl 14 | . 2 ⊢ (𝜑 → (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘∅) = 𝐶) |
6 | 2, 5 | syl5eq 2127 | 1 ⊢ (𝜑 → (𝐺‘∅) = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 ∈ wcel 1434 ∅c0 3269 ↦ cmpt 3865 ‘cfv 4968 (class class class)co 5590 freccfrec 6086 1c1 7253 + caddc 7255 ℤcz 8645 |
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 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-id 4083 df-iord 4156 df-on 4158 df-suc 4161 df-iom 4368 df-xp 4406 df-rel 4407 df-cnv 4408 df-co 4409 df-dm 4410 df-res 4412 df-iota 4933 df-fun 4970 df-fn 4971 df-fv 4976 df-recs 6001 df-frec 6087 |
This theorem is referenced by: frec2uzuzd 9697 frec2uzrand 9700 frec2uzrdg 9704 frecuzrdgg 9711 frecfzennn 9721 0tonninf 9733 1tonninf 9734 omgadd 10044 |
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