Theorem fssxp 6697

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 6675	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → Rel 𝐹)
2		relssdmrn 6235	. . 3 ⊢ (Rel 𝐹 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
3	1, 2	syl 17	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
4		fdm 6679	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
5		eqimss 3994	. . . 4 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴)
6	4, 5	syl 17	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 ⊆ 𝐴)
7		frn 6677	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
8		xpss12 5647	. . 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 3946	1 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))

Syntax hints:   → wi 4   = wceq 1542   ⊆ wss 3903   × cxp 5630  dom cdm 5632  ran crn 5633  Rel wrel 5637  ⟶wf 6496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-rn 5643  df-fun 6502  df-fn 6503  df-f 6504

This theorem is referenced by:  funssxp  6698  opelf  6703  dff2  7053  dff3  7054  fndifnfp  7132  fex2  7888  fabexd  7889  fabexgOLD  7891  f2ndf  8072  f1o2ndf1  8074  mapexOLD  8781  fsetsspwxp  8802  uniixp  8871  wdom2d  9497  rankfu  9801  dfac12lem2  10067  infmap2  10139  axdc3lem  10372  fnct  10459  tskcard  10704  ixxex  13284  imasvscafn  17470  imasvscaf  17472  fnmrc  17542  mrcfval  17543  isacs1i  17592  mreacs  17593  pjfval  21673  pjpm  21675  isngp2  24553  volf  25498  fgraphopab  43554  dfno2  43778  issmflem  47079