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Mirrors > Home > ILE Home > Th. List > fssxp | GIF version |
Description: A mapping is a class of ordered pairs. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
fssxp | ⊢ (𝐹:A⟶B → 𝐹 ⊆ (A × B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frel 4992 | . . 3 ⊢ (𝐹:A⟶B → Rel 𝐹) | |
2 | relssdmrn 4784 | . . 3 ⊢ (Rel 𝐹 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝐹:A⟶B → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) |
4 | fdm 4993 | . . . 4 ⊢ (𝐹:A⟶B → dom 𝐹 = A) | |
5 | eqimss 2991 | . . . 4 ⊢ (dom 𝐹 = A → dom 𝐹 ⊆ A) | |
6 | 4, 5 | syl 14 | . . 3 ⊢ (𝐹:A⟶B → dom 𝐹 ⊆ A) |
7 | frn 4995 | . . 3 ⊢ (𝐹:A⟶B → ran 𝐹 ⊆ B) | |
8 | xpss12 4388 | . . 3 ⊢ ((dom 𝐹 ⊆ A ∧ ran 𝐹 ⊆ B) → (dom 𝐹 × ran 𝐹) ⊆ (A × B)) | |
9 | 6, 7, 8 | syl2anc 391 | . 2 ⊢ (𝐹:A⟶B → (dom 𝐹 × ran 𝐹) ⊆ (A × B)) |
10 | 3, 9 | sstrd 2949 | 1 ⊢ (𝐹:A⟶B → 𝐹 ⊆ (A × B)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ⊆ wss 2911 × cxp 4286 dom cdm 4288 ran crn 4289 Rel wrel 4293 ⟶wf 4841 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-xp 4294 df-rel 4295 df-cnv 4296 df-dm 4298 df-rn 4299 df-fun 4847 df-fn 4848 df-f 4849 |
This theorem is referenced by: fex2 5002 funssxp 5003 opelf 5005 fabexg 5020 dff2 5254 dff3im 5255 f2ndf 5789 f1o2ndf1 5791 tfrlemibfn 5883 ixxex 8538 |
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