Theorem fssxp 5001

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 4992	. . 3 ⊢ (𝐹:A⟶B → Rel 𝐹)
2		relssdmrn 4784	. . 3 ⊢ (Rel 𝐹 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
3	1, 2	syl 14	. 2 ⊢ (𝐹:A⟶B → 𝐹 ⊆ (dom 𝐹 × ran 𝐹))
4		fdm 4993	. . . 4 ⊢ (𝐹:A⟶B → dom 𝐹 = A)
5		eqimss 2991	. . . 4 ⊢ (dom 𝐹 = A → dom 𝐹 ⊆ A)
6	4, 5	syl 14	. . 3 ⊢ (𝐹:A⟶B → dom 𝐹 ⊆ A)
7		frn 4995	. . 3 ⊢ (𝐹:A⟶B → ran 𝐹 ⊆ B)
8		xpss12 4388	. . 3 ⊢ ((dom 𝐹 ⊆ A ∧ ran 𝐹 ⊆ B) → (dom 𝐹 × ran 𝐹) ⊆ (A × B))
9	6, 7, 8	syl2anc 391	. 2 ⊢ (𝐹:A⟶B → (dom 𝐹 × ran 𝐹) ⊆ (A × B))
10	3, 9	sstrd 2949	1 ⊢ (𝐹:A⟶B → 𝐹 ⊆ (A × B))

Syntax hints:   → wi 4   = wceq 1242   ⊆ wss 2911   × cxp 4286  dom cdm 4288  ran crn 4289  Rel wrel 4293  ⟶wf 4841

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849

This theorem is referenced by:  fex2  5002  funssxp  5003  opelf  5005  fabexg  5020  dff2  5254  dff3im  5255  f2ndf  5789  f1o2ndf1  5791  tfrlemibfn  5883  ixxex  8498