Theorem fonex 49452

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 3062	. . 3 ⊢ ¬ 𝐵 ∈ V
3		fonex.2	. . . . . 6 ⊢ 𝐹:𝐴–onto→𝐵
4		fofun 6775	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹)
5	3, 4	ax-mp 5	. . . . 5 ⊢ Fun 𝐹
6		funrnex 7931	. . . . 5 ⊢ (dom 𝐹 ∈ V → (Fun 𝐹 → ran 𝐹 ∈ V))
7	5, 6	mpi 20	. . . 4 ⊢ (dom 𝐹 ∈ V → ran 𝐹 ∈ V)
8		fofn 6776	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
9	3, 8	ax-mp 5	. . . . . 6 ⊢ 𝐹 Fn 𝐴
10	9	fndmi 6621	. . . . 5 ⊢ dom 𝐹 = 𝐴
11	10	eleq1i 2852	. . . 4 ⊢ (dom 𝐹 ∈ V ↔ 𝐴 ∈ V)
12		forn 6777	. . . . . 6 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
13	3, 12	ax-mp 5	. . . . 5 ⊢ ran 𝐹 = 𝐵
14	13	eleq1i 2852	. . . 4 ⊢ (ran 𝐹 ∈ V ↔ 𝐵 ∈ V)
15	7, 11, 14	3imtr3i 293	. . 3 ⊢ (𝐴 ∈ V → 𝐵 ∈ V)
16	2, 15	mto 199	. 2 ⊢ ¬ 𝐴 ∈ V
17	16	nelir 3063	1 ⊢ 𝐴 ∉ V

Syntax hints:   = wceq 1559   ∈ wcel 2141   ∉ wnel 3060  Vcvv 3453  dom cdm 5645  ran crn 5646  Fun wfun 6511   Fn wfn 6512  –onto→wfo 6515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525

This theorem is referenced by:  posnex  49565  prsnex  49566  termcnex  50161