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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 6296 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → Rel 𝐹) | |
2 | relssdmrn 5910 | . . 3 ⊢ (Rel 𝐹 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) |
4 | fdm 6299 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
5 | eqimss 3875 | . . . 4 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 ⊆ 𝐴) |
7 | frn 6297 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
8 | xpss12 5370 | . . 3 ⊢ ((dom 𝐹 ⊆ 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → (dom 𝐹 × ran 𝐹) ⊆ (𝐴 × 𝐵)) | |
9 | 6, 7, 8 | syl2anc 579 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (dom 𝐹 × ran 𝐹) ⊆ (𝐴 × 𝐵)) |
10 | 3, 9 | sstrd 3830 | 1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ⊆ wss 3791 × cxp 5353 dom cdm 5355 ran crn 5356 Rel wrel 5360 ⟶wf 6131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4887 df-opab 4949 df-xp 5361 df-rel 5362 df-cnv 5363 df-dm 5365 df-rn 5366 df-fun 6137 df-fn 6138 df-f 6139 |
This theorem is referenced by: funssxp 6311 opelf 6315 dff2 6635 dff3 6636 fndifnfp 6709 fex2 7400 fabexg 7401 f2ndf 7564 f1o2ndf1 7566 mapex 8146 uniixp 8217 wdom2d 8774 rankfu 9037 dfac12lem2 9301 infmap2 9375 axdc3lem 9607 fnct 9694 tskcard 9938 ixxex 12498 imasvscafn 16583 imasvscaf 16585 fnmrc 16653 mrcfval 16654 isacs1i 16703 mreacs 16704 pjfval 20449 pjpm 20451 isngp2 22809 volf 23733 fgraphopab 38729 issmflem 41845 |
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