Theorem fornex 2793

Description: If the domain of an onto function exists, so does its codomain.

Assertion

Ref	Expression
fornex	⊢ (A ∈ C → (F:A–onto→B → B ∈ V))

Proof of Theorem fornex

Step	Hyp	Ref	Expression
1		fof 2788	. . . 4 ⊢ (F:A–onto→B → F:A–→B)
2		ffun 2754	. . . 4 ⊢ (F:A–→B → Fun F)
3		funrnex 2743	. . . . 5 ⊢ (dom F ∈ C → (Fun F → ran F ∈ V))
4	3	com12 13	. . . 4 ⊢ (Fun F → (dom F ∈ C → ran F ∈ V))
5	1, 2, 4	3syl 21	. . 3 ⊢ (F:A–onto→B → (dom F ∈ C → ran F ∈ V))
6		fdm 2756	. . . . 5 ⊢ (F:A–→B → dom F = A)
7	1, 6	syl 12	. . . 4 ⊢ (F:A–onto→B → dom F = A)
8	7	eleq1d 1155	. . 3 ⊢ (F:A–onto→B → (dom F ∈ C ↔ A ∈ C))
9		forn 2789	. . . 4 ⊢ (F:A–onto→B → ran F = B)
10	9	eleq1d 1155	. . 3 ⊢ (F:A–onto→B → (ran F ∈ V ↔ B ∈ V))
11	5, 8, 10	3imtr3d 420	. 2 ⊢ (F:A–onto→B → (A ∈ C → B ∈ V))
12	11	com12 13	1 ⊢ (A ∈ C → (F:A–onto→B → B ∈ V))

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092  Vcvv 1348  dom cdm 2410  ran crn 2411  Fun wfun 2416  –→wf 2418  –onto→wfo 2420

This theorem is referenced by:  f1dmex 2819  f1imaen 3327

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-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  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-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-fo 2436

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