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Mirrors > Home > MPE Home > Th. List > fssxp | Structured version Visualization version 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 6492 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → Rel 𝐹) | |
2 | relssdmrn 6088 | . . 3 ⊢ (Rel 𝐹 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) |
4 | fdm 6495 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
5 | eqimss 3971 | . . . 4 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 ⊆ 𝐴) |
7 | frn 6493 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
8 | xpss12 5534 | . . 3 ⊢ ((dom 𝐹 ⊆ 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → (dom 𝐹 × ran 𝐹) ⊆ (𝐴 × 𝐵)) | |
9 | 6, 7, 8 | syl2anc 587 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (dom 𝐹 × ran 𝐹) ⊆ (𝐴 × 𝐵)) |
10 | 3, 9 | sstrd 3925 | 1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ⊆ wss 3881 × cxp 5517 dom cdm 5519 ran crn 5520 Rel wrel 5524 ⟶wf 6320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-fun 6326 df-fn 6327 df-f 6328 |
This theorem is referenced by: funssxp 6509 opelf 6513 dff2 6842 dff3 6843 fndifnfp 6915 fex2 7620 fabexg 7621 f2ndf 7799 f1o2ndf1 7801 mapex 8395 uniixp 8468 wdom2d 9028 rankfu 9290 dfac12lem2 9555 infmap2 9629 axdc3lem 9861 fnct 9948 tskcard 10192 ixxex 12737 imasvscafn 16802 imasvscaf 16804 fnmrc 16870 mrcfval 16871 isacs1i 16920 mreacs 16921 pjfval 20395 pjpm 20397 isngp2 23203 volf 24133 fgraphopab 40154 issmflem 43361 |
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