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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 5530 | . 2 ⊢ (𝐺‘∅) = (frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)‘∅) |
3 | frec2uz.1 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℤ) | |
4 | frec0g 6415 | . . 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 | eqtrid 2233 | 1 ⊢ (𝜑 → (𝐺‘∅) = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1363 ∈ wcel 2159 ∅c0 3436 ↦ cmpt 4078 ‘cfv 5230 (class class class)co 5890 freccfrec 6408 1c1 7829 + caddc 7831 ℤcz 9270 |
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 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2161 ax-14 2162 ax-ext 2170 ax-sep 4135 ax-nul 4143 ax-pow 4188 ax-pr 4223 ax-un 4447 ax-setind 4550 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2040 df-mo 2041 df-clab 2175 df-cleq 2181 df-clel 2184 df-nfc 2320 df-ne 2360 df-ral 2472 df-rex 2473 df-rab 2476 df-v 2753 df-sbc 2977 df-csb 3072 df-dif 3145 df-un 3147 df-in 3149 df-ss 3156 df-nul 3437 df-pw 3591 df-sn 3612 df-pr 3613 df-op 3615 df-uni 3824 df-int 3859 df-iun 3902 df-br 4018 df-opab 4079 df-mpt 4080 df-tr 4116 df-id 4307 df-iord 4380 df-on 4382 df-suc 4385 df-iom 4604 df-xp 4646 df-rel 4647 df-cnv 4648 df-co 4649 df-dm 4650 df-res 4652 df-iota 5192 df-fun 5232 df-fn 5233 df-fv 5238 df-recs 6323 df-frec 6409 |
This theorem is referenced by: frec2uzuzd 10419 frec2uzrand 10422 frec2uzrdg 10426 frecuzrdgg 10433 frecfzennn 10443 0tonninf 10456 1tonninf 10457 omgadd 10799 ennnfonelem1 12425 ennnfonelemhf1o 12431 012of 15129 2o01f 15130 isomninnlem 15162 iswomninnlem 15181 ismkvnnlem 15184 |
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