Theorem slotresfo 50106

Description: The condition of a structure component extractor restricted to a class being a surjection. This combined with fonex 50071 can be used to prove a class being proper. (Contributed by Zhi Wang, 20-Oct-2025.)

Hypotheses

Ref	Expression
slotresfo.e	⊢ 𝐸 Fn V
slotresfo.v	⊢ (𝑘 ∈ 𝐴 → (𝐸‘𝑘) ∈ 𝑉)
slotresfo.k	⊢ (𝑏 ∈ 𝑉 → 𝐾 ∈ 𝐴)
slotresfo.b	⊢ (𝑏 ∈ 𝑉 → 𝑏 = (𝐸‘𝐾))

Assertion

Ref	Expression
slotresfo	⊢ (𝐸 ↾ 𝐴):𝐴–onto→𝑉

Distinct variable groups:   𝐴,𝑏,𝑘   𝐸,𝑏,𝑘   𝑘,𝐾   𝑉,𝑏,𝑘

Allowed substitution hint:   𝐾(𝑏)

Proof of Theorem slotresfo

Step	Hyp	Ref	Expression
1		slotresfo.e	. . . 4 ⊢ 𝐸 Fn V
2		ssv 3932	. . . 4 ⊢ 𝐴 ⊆ V
3		fnssres 6662	. . . 4 ⊢ ((𝐸 Fn V ∧ 𝐴 ⊆ V) → (𝐸 ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 705	. . 3 ⊢ (𝐸 ↾ 𝐴) Fn 𝐴
5		fvres 6904	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 → ((𝐸 ↾ 𝐴)‘𝑘) = (𝐸‘𝑘))
6		slotresfo.v	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 → (𝐸‘𝑘) ∈ 𝑉)
7	5, 6	eqeltrd 2860	. . . . 5 ⊢ (𝑘 ∈ 𝐴 → ((𝐸 ↾ 𝐴)‘𝑘) ∈ 𝑉)
8	7	rgen 3078	. . . 4 ⊢ ∀𝑘 ∈ 𝐴 ((𝐸 ↾ 𝐴)‘𝑘) ∈ 𝑉
9		fnfvrnss 7121	. . . 4 ⊢ (((𝐸 ↾ 𝐴) Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 ((𝐸 ↾ 𝐴)‘𝑘) ∈ 𝑉) → ran (𝐸 ↾ 𝐴) ⊆ 𝑉)
10	4, 8, 9	mp2an 705	. . 3 ⊢ ran (𝐸 ↾ 𝐴) ⊆ 𝑉
11		df-f 6542	. . 3 ⊢ ((𝐸 ↾ 𝐴):𝐴⟶𝑉 ↔ ((𝐸 ↾ 𝐴) Fn 𝐴 ∧ ran (𝐸 ↾ 𝐴) ⊆ 𝑉))
12	4, 10, 11	mpbir2an 724	. 2 ⊢ (𝐸 ↾ 𝐴):𝐴⟶𝑉
13		fveq2 6885	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝐸‘𝑘) = (𝐸‘𝐾))
14	13	eqeq2d 2771	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑏 = (𝐸‘𝑘) ↔ 𝑏 = (𝐸‘𝐾)))
15		slotresfo.k	. . . . 5 ⊢ (𝑏 ∈ 𝑉 → 𝐾 ∈ 𝐴)
16		slotresfo.b	. . . . 5 ⊢ (𝑏 ∈ 𝑉 → 𝑏 = (𝐸‘𝐾))
17	14, 15, 16	rspcedvdw 3579	. . . 4 ⊢ (𝑏 ∈ 𝑉 → ∃𝑘 ∈ 𝐴 𝑏 = (𝐸‘𝑘))
18	5	eqeq2d 2771	. . . . 5 ⊢ (𝑘 ∈ 𝐴 → (𝑏 = ((𝐸 ↾ 𝐴)‘𝑘) ↔ 𝑏 = (𝐸‘𝑘)))
19	18	rexbiia 3107	. . . 4 ⊢ (∃𝑘 ∈ 𝐴 𝑏 = ((𝐸 ↾ 𝐴)‘𝑘) ↔ ∃𝑘 ∈ 𝐴 𝑏 = (𝐸‘𝑘))
20	17, 19	sylibr 237	. . 3 ⊢ (𝑏 ∈ 𝑉 → ∃𝑘 ∈ 𝐴 𝑏 = ((𝐸 ↾ 𝐴)‘𝑘))
21	20	rgen 3078	. 2 ⊢ ∀𝑏 ∈ 𝑉 ∃𝑘 ∈ 𝐴 𝑏 = ((𝐸 ↾ 𝐴)‘𝑘)
22		dffo3 7102	. 2 ⊢ ((𝐸 ↾ 𝐴):𝐴–onto→𝑉 ↔ ((𝐸 ↾ 𝐴):𝐴⟶𝑉 ∧ ∀𝑏 ∈ 𝑉 ∃𝑘 ∈ 𝐴 𝑏 = ((𝐸 ↾ 𝐴)‘𝑘)))
23	12, 21, 22	mpbir2an 724	1 ⊢ (𝐸 ↾ 𝐴):𝐴–onto→𝑉

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895  ran crn 5651   ↾ cres 5652   Fn wfn 6533  ⟶wf 6534  –onto→wfo 6536  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546

This theorem is used by:  basresprsfo  50186  basrestermcfo  50806