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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 5080 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → Rel 𝐹) | |
2 | relssdmrn 4871 | . . 3 ⊢ (Rel 𝐹 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) |
4 | fdm 5081 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
5 | eqimss 3052 | . . . 4 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴) | |
6 | 4, 5 | syl 14 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 ⊆ 𝐴) |
7 | frn 5083 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
8 | xpss12 4473 | . . 3 ⊢ ((dom 𝐹 ⊆ 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → (dom 𝐹 × ran 𝐹) ⊆ (𝐴 × 𝐵)) | |
9 | 6, 7, 8 | syl2anc 403 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (dom 𝐹 × ran 𝐹) ⊆ (𝐴 × 𝐵)) |
10 | 3, 9 | sstrd 3010 | 1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 ⊆ wss 2974 × cxp 4369 dom cdm 4371 ran crn 4372 Rel wrel 4376 ⟶wf 4928 |
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-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-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-sep 3904 ax-pow 3956 ax-pr 3972 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ral 2354 df-rex 2355 df-v 2604 df-un 2978 df-in 2980 df-ss 2987 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-br 3794 df-opab 3848 df-xp 4377 df-rel 4378 df-cnv 4379 df-dm 4381 df-rn 4382 df-fun 4934 df-fn 4935 df-f 4936 |
This theorem is referenced by: fex2 5090 funssxp 5091 opelf 5093 fabexg 5108 dff2 5343 dff3im 5344 f2ndf 5878 f1o2ndf1 5880 tfrlemibfn 5977 tfr1onlembfn 5993 tfrcllembfn 6006 ixxex 8998 |
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