Theorem fonex 48774

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 3047	. . 3 ⊢ ¬ 𝐵 ∈ V
3		fonex.2	. . . . . 6 ⊢ 𝐹:𝐴–onto→𝐵
4		fofun 6820	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹)
5	3, 4	ax-mp 5	. . . . 5 ⊢ Fun 𝐹
6		funrnex 7979	. . . . 5 ⊢ (dom 𝐹 ∈ V → (Fun 𝐹 → ran 𝐹 ∈ V))
7	5, 6	mpi 20	. . . 4 ⊢ (dom 𝐹 ∈ V → ran 𝐹 ∈ V)
8		fofn 6821	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
9	3, 8	ax-mp 5	. . . . . 6 ⊢ 𝐹 Fn 𝐴
10	9	fndmi 6671	. . . . 5 ⊢ dom 𝐹 = 𝐴
11	10	eleq1i 2831	. . . 4 ⊢ (dom 𝐹 ∈ V ↔ 𝐴 ∈ V)
12		forn 6822	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
13	3, 12	ax-mp 5	. . . . 5 ⊢ ran 𝐹 = 𝐵
14	13	eleq1i 2831	. . . 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 3048	1 ⊢ 𝐴 ∉ V

Syntax hints:   = wceq 1539   ∈ wcel 2107   ∉ wnel 3045  Vcvv 3479  dom cdm 5684  ran crn 5685  Fun wfun 6554   Fn wfn 6555  –onto→wfo 6558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568

This theorem is referenced by:  posnex  48884  prsnex  48885  termcnex  49228