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Theorem fssxp 6628

Description: A mapping is a class of ordered pairs. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)

**Assertion**
Ref	Expression
fssxp	⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))

**Proof of Theorem fssxp**
Step	Hyp	Ref	Expression
1		frel 6605	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → Rel 𝐹)
2		relssdmrn 6172	. . 3 ⊢ (Rel 𝐹 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
3	1, 2	syl 17	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
4		fdm 6609	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
5		eqimss 3977	. . . 4 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴)
6	4, 5	syl 17	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 ⊆ 𝐴)
7		frn 6607	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
8		xpss12 5604	. . 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 3931	1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ⊆ wss 3887 × cxp 5587 dom cdm 5589 ran crn 5590 Rel wrel 5594 ⟶wf 6429

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-xp 5595 df-rel 5596 df-cnv 5597 df-dm 5599 df-rn 5600 df-fun 6435 df-fn 6436 df-f 6437

This theorem is referenced by: funssxp 6629 opelf 6635 dff2 6975 dff3 6976 fndifnfp 7048 fex2 7780 fabexg 7781 f2ndf 7961 f1o2ndf1 7963 mapex 8621 fsetsspwxp 8641 uniixp 8709 wdom2d 9339 rankfu 9635 dfac12lem2 9900 infmap2 9974 axdc3lem 10206 fnct 10293 tskcard 10537 ixxex 13090 imasvscafn 17248 imasvscaf 17250 fnmrc 17316 mrcfval 17317 isacs1i 17366 mreacs 17367 pjfval 20913 pjpm 20915 isngp2 23753 volf 24693 fgraphopab 41035 issmflem 44263