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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 6572 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → Rel 𝐹) | |
2 | relssdmrn 6150 | . . 3 ⊢ (Rel 𝐹 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) |
4 | fdm 6576 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
5 | eqimss 3974 | . . . 4 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 ⊆ 𝐴) |
7 | frn 6574 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) | |
8 | xpss12 5584 | . . 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 3928 | 1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ⊆ wss 3883 × cxp 5567 dom cdm 5569 ran crn 5570 Rel wrel 5574 ⟶wf 6397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-12 2177 ax-ext 2710 ax-sep 5209 ax-nul 5216 ax-pr 5339 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2717 df-cleq 2731 df-clel 2818 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3425 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4255 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-br 5071 df-opab 5133 df-xp 5575 df-rel 5576 df-cnv 5577 df-dm 5579 df-rn 5580 df-fun 6403 df-fn 6404 df-f 6405 |
This theorem is referenced by: funssxp 6596 opelf 6602 dff2 6940 dff3 6941 fndifnfp 7013 fex2 7733 fabexg 7734 f2ndf 7911 f1o2ndf1 7913 mapex 8538 fsetsspwxp 8558 uniixp 8626 wdom2d 9226 rankfu 9523 dfac12lem2 9788 infmap2 9862 axdc3lem 10094 fnct 10181 tskcard 10425 ixxex 12976 imasvscafn 17075 imasvscaf 17077 fnmrc 17143 mrcfval 17144 isacs1i 17193 mreacs 17194 pjfval 20701 pjpm 20703 isngp2 23527 volf 24458 fgraphopab 40786 issmflem 43981 |
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