Theorem fssxp 4983

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 4975	. . 3 ⊢ (𝐹:A⟶B → Rel 𝐹)
2		relssdmrn 4768	. . 3 ⊢ (Rel 𝐹 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
3	1, 2	syl 14	. 2 ⊢ (𝐹:A⟶B → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
4		fdm 4976	. . . 4 ⊢ (𝐹:A⟶B → dom 𝐹 = A)
5		eqimss 2974	. . . 4 ⊢ (dom 𝐹 = A → dom 𝐹 ⊆ A)
6	4, 5	syl 14	. . 3 ⊢ (𝐹:A⟶B → dom 𝐹 ⊆ A)
7		frn 4978	. . 3 ⊢ (𝐹:A⟶B → ran 𝐹 ⊆ B)
8		xpss12 4372	. . 3 ⊢ ((dom 𝐹 ⊆ A ∧ ran 𝐹 ⊆ B) → (dom 𝐹 × ran 𝐹) ⊆ (A × B))
9	6, 7, 8	syl2anc 393	. 2 ⊢ (𝐹:A⟶B → (dom 𝐹 × ran 𝐹) ⊆ (A × B))
10	3, 9	sstrd 2932	1 ⊢ (𝐹:A⟶B → 𝐹 ⊆ (A × B))

Syntax hints:   → wi 4   = wceq 1228   ⊆ wss 2894   × cxp 4270  dom cdm 4272  ran crn 4273  Rel wrel 4277  ⟶wf 4825

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-rel 4279  df-cnv 4280  df-dm 4282  df-rn 4283  df-fun 4831  df-fn 4832  df-f 4833

This theorem is referenced by:  fex2  4984  funssxp  4985  opelf  4987  fabexg  5002  dff2  5236  dff3im  5237  f2ndf  5770  f1o2ndf1  5772  tfrlemibfn  5863