Theorem foima 5561

Description: The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)

Assertion

Ref	Expression
foima	⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵)

Proof of Theorem foima

Step	Hyp	Ref	Expression
1		imadmrn 5084	. 2 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
2		fof 5556	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵)
3		fdm 5485	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
4	2, 3	syl 14	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → dom 𝐹 = 𝐴)
5	4	imaeq2d 5074	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴))
6		forn 5559	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
7	1, 5, 6	3eqtr3a 2286	1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵)

Syntax hints:   → wi 4   = wceq 1395  dom cdm 4723  ran crn 4724   “ cima 4726  ⟶wf 5320  –onto→wfo 5322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-xp 4729  df-cnv 4731  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-fn 5327  df-f 5328  df-fo 5330

This theorem is referenced by:  foimacnv  5598  foima2  5887  fiintim  7116  fidcenumlemr  7145