Theorem slotresfo 50228

Description: The condition of a structure component extractor restricted to a class being a surjection. This combined with fonex 50193 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⟶onto𝑉

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 3937	. . . 4 ⊢ 𝐴 ⊆ V
3		fnssres 6687	. . . 4 ⊢ ((𝐸 Fn V ∧ 𝐴 ⊆ V) → (𝐸 ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 705	. . 3 ⊢ (𝐸 ↾ 𝐴) Fn 𝐴
5		fvres 6930	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 → ((𝐸 ↾ 𝐴)‘𝑘) = (𝐸‘𝑘))
6		slotresfo.v	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 → (𝐸‘𝑘) ∈ 𝑉)
7	5, 6	eqeltrd 2864	. . . . 5 ⊢ (𝑘 ∈ 𝐴 → ((𝐸 ↾ 𝐴)‘𝑘) ∈ 𝑉)
8	7	rgen 3082	. . . 4 ⊢ ∀𝑘 ∈ 𝐴 ((𝐸 ↾ 𝐴)‘𝑘) ∈ 𝑉
9		fnfvrnss 7148	. . . 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 6565	. . 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 6911	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝐸‘𝑘) = (𝐸‘𝐾))
14	13	eqeq2d 2774	. . . . 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 3583	. . . 4 ⊢ (𝑏 ∈ 𝑉 → ∃𝑘 ∈ 𝐴 𝑏 = (𝐸‘𝑘))
18	5	eqeq2d 2774	. . . . 5 ⊢ (𝑘 ∈ 𝐴 → (𝑏 = ((𝐸 ↾ 𝐴)‘𝑘) ↔ 𝑏 = (𝐸‘𝑘)))
19	18	rexbiia 3111	. . . 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 3082	. 2 ⊢ ∀𝑏 ∈ 𝑉 ∃𝑘 ∈ 𝐴 𝑏 = ((𝐸 ↾ 𝐴)‘𝑘)
22		dffo3 7129	. 2 ⊢ ((𝐸 ↾ 𝐴):𝐴⟶onto𝑉 ↔ ((𝐸 ↾ 𝐴):𝐴⟶𝑉 ∧ ∀𝑏 ∈ 𝑉 ∃𝑘 ∈ 𝐴 𝑏 = ((𝐸 ↾ 𝐴)‘𝑘)))
23	12, 21, 22	mpbir2an 724	1 ⊢ (𝐸 ↾ 𝐴):𝐴⟶onto𝑉

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900  ran crn 5657   ↾ cres 5658   Fn wfn 6556  ⟶wf 6557  ⟶ontowfo 6559  ‘cfv 6561

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569

This theorem is used by:  basresprsfo  50308  basrestermcfo  50928