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Mirrors > Home > ILE Home > Th. List > omfnex | GIF version |
Description: The characteristic function for ordinal multiplication is defined everywhere. (Contributed by Jim Kingdon, 23-Aug-2019.) |
Ref | Expression |
---|---|
omfnex | ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) Fn V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2613 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | oaexg 6112 | . . . 4 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑥 +𝑜 𝐴) ∈ V) | |
3 | 1, 2 | mpan 415 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 +𝑜 𝐴) ∈ V) |
4 | 3 | ralrimivw 2440 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ V (𝑥 +𝑜 𝐴) ∈ V) |
5 | eqid 2083 | . . 3 ⊢ (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) = (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) | |
6 | 5 | fnmpt 5076 | . 2 ⊢ (∀𝑥 ∈ V (𝑥 +𝑜 𝐴) ∈ V → (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) Fn V) |
7 | 4, 6 | syl 14 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ V ↦ (𝑥 +𝑜 𝐴)) Fn V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1434 ∀wral 2353 Vcvv 2610 ↦ cmpt 3859 Fn wfn 4947 (class class class)co 5563 +𝑜 coa 6082 |
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-coll 3913 ax-sep 3916 ax-pow 3968 ax-pr 3992 ax-un 4216 ax-setind 4308 |
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-reu 2360 df-rab 2362 df-v 2612 df-sbc 2825 df-csb 2918 df-dif 2984 df-un 2986 df-in 2988 df-ss 2995 df-nul 3268 df-pw 3402 df-sn 3422 df-pr 3423 df-op 3425 df-uni 3622 df-iun 3700 df-br 3806 df-opab 3860 df-mpt 3861 df-tr 3896 df-id 4076 df-iord 4149 df-on 4151 df-suc 4154 df-xp 4397 df-rel 4398 df-cnv 4399 df-co 4400 df-dm 4401 df-rn 4402 df-res 4403 df-ima 4404 df-iota 4917 df-fun 4954 df-fn 4955 df-f 4956 df-f1 4957 df-fo 4958 df-f1o 4959 df-fv 4960 df-ov 5566 df-oprab 5567 df-mpt2 5568 df-1st 5818 df-2nd 5819 df-recs 5974 df-irdg 6039 df-oadd 6089 |
This theorem is referenced by: fnom 6114 omexg 6115 omv 6119 omv2 6129 |
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