Theorem foima 2790

Description: The image of the domain of an onto function.

Assertion

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

Proof of Theorem foima

Step	Hyp	Ref	Expression
1		fof 2788	. . . 4 ⊢ (F:A–onto→B → F:A–→B)
2		fdm 2756	. . . 4 ⊢ (F:A–→B → dom F = A)
3		imaeq2 2603	. . . 4 ⊢ (dom F = A → (F “ dom F) = (F “ A))
4	1, 2, 3	3syl 21	. . 3 ⊢ (F:A–onto→B → (F “ dom F) = (F “ A))
5		imadmrn 2610	. . 3 ⊢ (F “ dom F) = ran F
6	4, 5	syl5reqr 1139	. 2 ⊢ (F:A–onto→B → (F “ A) = ran F)
7		forn 2789	. 2 ⊢ (F:A–onto→B → ran F = B)
8	6, 7	eqtrd 1128	1 ⊢ (F:A–onto→B → (F “ A) = B)

Syntax hints:   → wi 2   = wceq 1091  dom cdm 2410  ran crn 2411   “ cima 2413  –→wf 2418  –onto→wfo 2420

This theorem is referenced by:  fiint 3445

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fn 2433  df-f 2434  df-fo 2436

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