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| 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 6740 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → Rel 𝐹) | |
| 2 | relssdmrn 6287 | . . 3 ⊢ (Rel 𝐹 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) | |
| 3 | 1, 2 | syl 17 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) | 
| 4 | fdm 6744 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
| 5 | eqimss 4041 | . . . 4 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴) | |
| 6 | 4, 5 | syl 17 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 ⊆ 𝐴) | 
| 7 | frn 6742 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
| 8 | xpss12 5699 | . . 3 ⊢ ((dom 𝐹 ⊆ 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → (dom 𝐹 × ran 𝐹) ⊆ (𝐴 × 𝐵)) | |
| 9 | 6, 7, 8 | syl2anc 584 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (dom 𝐹 × ran 𝐹) ⊆ (𝐴 × 𝐵)) | 
| 10 | 3, 9 | sstrd 3993 | 1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ⊆ wss 3950 × cxp 5682 dom cdm 5684 ran crn 5685 Rel wrel 5689 ⟶wf 6556 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-cnv 5692 df-dm 5694 df-rn 5695 df-fun 6562 df-fn 6563 df-f 6564 | 
| This theorem is referenced by: funssxp 6763 opelf 6768 dff2 7118 dff3 7119 fndifnfp 7197 fex2 7959 fabexd 7960 fabexgOLD 7962 f2ndf 8146 f1o2ndf1 8148 mapexOLD 8873 fsetsspwxp 8894 uniixp 8962 wdom2d 9621 rankfu 9918 dfac12lem2 10186 infmap2 10258 axdc3lem 10491 fnct 10578 tskcard 10822 ixxex 13399 imasvscafn 17583 imasvscaf 17585 fnmrc 17651 mrcfval 17652 isacs1i 17701 mreacs 17702 pjfval 21727 pjpm 21729 isngp2 24611 volf 25565 fgraphopab 43220 dfno2 43446 issmflem 46747 | 
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