Theorem fonex 49108

Description: The domain of a surjection is a proper class if the range is a proper class as well. Can be used to prove that if a structure component extractor restricted to a class maps onto a proper class, then the class is a proper class as well. (Contributed by Zhi Wang, 20-Oct-2025.)

Hypotheses

Ref	Expression
fonex.1	⊢ 𝐵 ∉ V
fonex.2	⊢ 𝐹:𝐴–onto→𝐵

Assertion

Ref	Expression
fonex	⊢ 𝐴 ∉ V

Proof of Theorem fonex

Step	Hyp	Ref	Expression
1		fonex.1	. . . 4 ⊢ 𝐵 ∉ V
2	1	neli 3038	. . 3 ⊢ ¬ 𝐵 ∈ V
3		fonex.2	. . . . . 6 ⊢ 𝐹:𝐴–onto→𝐵
4		fofun 6747	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹)
5	3, 4	ax-mp 5	. . . . 5 ⊢ Fun 𝐹
6		funrnex 7898	. . . . 5 ⊢ (dom 𝐹 ∈ V → (Fun 𝐹 → ran 𝐹 ∈ V))
7	5, 6	mpi 20	. . . 4 ⊢ (dom 𝐹 ∈ V → ran 𝐹 ∈ V)
8		fofn 6748	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
9	3, 8	ax-mp 5	. . . . . 6 ⊢ 𝐹 Fn 𝐴
10	9	fndmi 6596	. . . . 5 ⊢ dom 𝐹 = 𝐴
11	10	eleq1i 2827	. . . 4 ⊢ (dom 𝐹 ∈ V ↔ 𝐴 ∈ V)
12		forn 6749	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
13	3, 12	ax-mp 5	. . . . 5 ⊢ ran 𝐹 = 𝐵
14	13	eleq1i 2827	. . . 4 ⊢ (ran 𝐹 ∈ V ↔ 𝐵 ∈ V)
15	7, 11, 14	3imtr3i 291	. . 3 ⊢ (𝐴 ∈ V → 𝐵 ∈ V)
16	2, 15	mto 197	. 2 ⊢ ¬ 𝐴 ∈ V
17	16	nelir 3039	1 ⊢ 𝐴 ∉ V

Syntax hints:   = wceq 1541   ∈ wcel 2113   ∉ wnel 3036  Vcvv 3440  dom cdm 5624  ran crn 5625  Fun wfun 6486   Fn wfn 6487  –onto→wfo 6490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500

This theorem is referenced by:  posnex  49221  prsnex  49222  termcnex  49817