Theorem fonex 48852

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 3031	. . 3 ⊢ ¬ 𝐵 ∈ V
3		fonex.2	. . . . . 6 ⊢ 𝐹:𝐴–onto→𝐵
4		fofun 6773	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹)
5	3, 4	ax-mp 5	. . . . 5 ⊢ Fun 𝐹
6		funrnex 7932	. . . . 5 ⊢ (dom 𝐹 ∈ V → (Fun 𝐹 → ran 𝐹 ∈ V))
7	5, 6	mpi 20	. . . 4 ⊢ (dom 𝐹 ∈ V → ran 𝐹 ∈ V)
8		fofn 6774	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
9	3, 8	ax-mp 5	. . . . . 6 ⊢ 𝐹 Fn 𝐴
10	9	fndmi 6622	. . . . 5 ⊢ dom 𝐹 = 𝐴
11	10	eleq1i 2819	. . . 4 ⊢ (dom 𝐹 ∈ V ↔ 𝐴 ∈ V)
12		forn 6775	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
13	3, 12	ax-mp 5	. . . . 5 ⊢ ran 𝐹 = 𝐵
14	13	eleq1i 2819	. . . 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 3032	1 ⊢ 𝐴 ∉ V

Syntax hints:   = wceq 1540   ∈ wcel 2109   ∉ wnel 3029  Vcvv 3447  dom cdm 5638  ran crn 5639  Fun wfun 6505   Fn wfn 6506  –onto→wfo 6509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519

This theorem is referenced by:  posnex  48965  prsnex  48966  termcnex  49562