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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 2684 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | oaexg 6337 | . . . 4 ⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ 𝑉) → (𝑥 +o 𝐴) ∈ V) | |
3 | 1, 2 | mpan 420 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 +o 𝐴) ∈ V) |
4 | 3 | ralrimivw 2504 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∀𝑥 ∈ V (𝑥 +o 𝐴) ∈ V) |
5 | eqid 2137 | . . 3 ⊢ (𝑥 ∈ V ↦ (𝑥 +o 𝐴)) = (𝑥 ∈ V ↦ (𝑥 +o 𝐴)) | |
6 | 5 | fnmpt 5244 | . 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 1480 ∀wral 2414 Vcvv 2681 ↦ cmpt 3984 Fn wfn 5113 (class class class)co 5767 +o coa 6303 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-oadd 6310 |
This theorem is referenced by: fnom 6339 omexg 6340 omv 6344 omv2 6354 |
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