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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 6678 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → Rel 𝐹) | |
2 | relssdmrn 6225 | . . 3 ⊢ (Rel 𝐹 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) |
4 | fdm 6682 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
5 | eqimss 4005 | . . . 4 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 ⊆ 𝐴) |
7 | frn 6680 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
8 | xpss12 5653 | . . 3 ⊢ ((dom 𝐹 ⊆ 𝐴 ∧ ran 𝐹 ⊆ 𝐵) → (dom 𝐹 × ran 𝐹) ⊆ (𝐴 × 𝐵)) | |
9 | 6, 7, 8 | syl2anc 585 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (dom 𝐹 × ran 𝐹) ⊆ (𝐴 × 𝐵)) |
10 | 3, 9 | sstrd 3959 | 1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ⊆ wss 3915 × cxp 5636 dom cdm 5638 ran crn 5639 Rel wrel 5643 ⟶wf 6497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-xp 5644 df-rel 5645 df-cnv 5646 df-dm 5648 df-rn 5649 df-fun 6503 df-fn 6504 df-f 6505 |
This theorem is referenced by: funssxp 6702 opelf 6708 dff2 7054 dff3 7055 fndifnfp 7127 fex2 7875 fabexg 7876 f2ndf 8057 f1o2ndf1 8059 mapex 8778 fsetsspwxp 8798 uniixp 8866 wdom2d 9523 rankfu 9820 dfac12lem2 10087 infmap2 10161 axdc3lem 10393 fnct 10480 tskcard 10724 ixxex 13282 imasvscafn 17426 imasvscaf 17428 fnmrc 17494 mrcfval 17495 isacs1i 17544 mreacs 17545 pjfval 21128 pjpm 21130 isngp2 23969 volf 24909 fgraphopab 41566 dfno2 41774 issmflem 45042 |
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