Theorem fonex 48736

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 3037	. . 3 ⊢ ¬ 𝐵 ∈ V
3		fonex.2	. . . . . 6 ⊢ 𝐹:𝐴–onto→𝐵
4		fofun 6788	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹)
5	3, 4	ax-mp 5	. . . . 5 ⊢ Fun 𝐹
6		funrnex 7947	. . . . 5 ⊢ (dom 𝐹 ∈ V → (Fun 𝐹 → ran 𝐹 ∈ V))
7	5, 6	mpi 20	. . . 4 ⊢ (dom 𝐹 ∈ V → ran 𝐹 ∈ V)
8		fofn 6789	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
9	3, 8	ax-mp 5	. . . . . 6 ⊢ 𝐹 Fn 𝐴
10	9	fndmi 6639	. . . . 5 ⊢ dom 𝐹 = 𝐴
11	10	eleq1i 2824	. . . 4 ⊢ (dom 𝐹 ∈ V ↔ 𝐴 ∈ V)
12		forn 6790	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
13	3, 12	ax-mp 5	. . . . 5 ⊢ ran 𝐹 = 𝐵
14	13	eleq1i 2824	. . . 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 3038	1 ⊢ 𝐴 ∉ V

Syntax hints:   = wceq 1539   ∈ wcel 2107   ∉ wnel 3035  Vcvv 3457  dom cdm 5652  ran crn 5653  Fun wfun 6522   Fn wfn 6523  –onto→wfo 6526

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 2706  ax-rep 5247  ax-sep 5264  ax-nul 5274  ax-pr 5400  ax-un 7724

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 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-iun 4967  df-br 5118  df-opab 5180  df-mpt 5200  df-id 5546  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536

This theorem is referenced by:  posnex  48848  prsnex  48849  termcnex  49314