Theorem slotresfo 49528

Description: The condition of a structure component extractor restricted to a class being a surjection. This combined with fonex 49496 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 3963	. . . 4 ⊢ 𝐴 ⊆ V
3		fnssres 6648	. . . 4 ⊢ ((𝐸 Fn V ∧ 𝐴 ⊆ V) → (𝐸 ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 704	. . 3 ⊢ (𝐸 ↾ 𝐴) Fn 𝐴
5		fvres 6890	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 → ((𝐸 ↾ 𝐴)‘𝑘) = (𝐸‘𝑘))
6		slotresfo.v	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 → (𝐸‘𝑘) ∈ 𝑉)
7	5, 6	eqeltrd 2865	. . . . 5 ⊢ (𝑘 ∈ 𝐴 → ((𝐸 ↾ 𝐴)‘𝑘) ∈ 𝑉)
8	7	rgen 3081	. . . 4 ⊢ ∀𝑘 ∈ 𝐴 ((𝐸 ↾ 𝐴)‘𝑘) ∈ 𝑉
9		fnfvrnss 7106	. . . 4 ⊢ (((𝐸 ↾ 𝐴) Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 ((𝐸 ↾ 𝐴)‘𝑘) ∈ 𝑉) → ran (𝐸 ↾ 𝐴) ⊆ 𝑉)
10	4, 8, 9	mp2an 704	. . 3 ⊢ ran (𝐸 ↾ 𝐴) ⊆ 𝑉
11		df-f 6529	. . 3 ⊢ ((𝐸 ↾ 𝐴):𝐴⟶𝑉 ↔ ((𝐸 ↾ 𝐴) Fn 𝐴 ∧ ran (𝐸 ↾ 𝐴) ⊆ 𝑉))
12	4, 10, 11	mpbir2an 723	. 2 ⊢ (𝐸 ↾ 𝐴):𝐴⟶𝑉
13		fveq2 6871	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝐸‘𝑘) = (𝐸‘𝐾))
14	13	eqeq2d 2776	. . . . 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 3587	. . . 4 ⊢ (𝑏 ∈ 𝑉 → ∃𝑘 ∈ 𝐴 𝑏 = (𝐸‘𝑘))
18	5	eqeq2d 2776	. . . . 5 ⊢ (𝑘 ∈ 𝐴 → (𝑏 = ((𝐸 ↾ 𝐴)‘𝑘) ↔ 𝑏 = (𝐸‘𝑘)))
19	18	rexbiia 3110	. . . 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 3081	. 2 ⊢ ∀𝑏 ∈ 𝑉 ∃𝑘 ∈ 𝐴 𝑏 = ((𝐸 ↾ 𝐴)‘𝑘)
22		dffo3 7087	. 2 ⊢ ((𝐸 ↾ 𝐴):𝐴–onto→𝑉 ↔ ((𝐸 ↾ 𝐴):𝐴⟶𝑉 ∧ ∀𝑏 ∈ 𝑉 ∃𝑘 ∈ 𝐴 𝑏 = ((𝐸 ↾ 𝐴)‘𝑘)))
23	12, 21, 22	mpbir2an 723	1 ⊢ (𝐸 ↾ 𝐴):𝐴–onto→𝑉

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089  Vcvv 3457   ⊆ wss 3907  ran crn 5653   ↾ cres 5654   Fn wfn 6520  ⟶wf 6521  –onto→wfo 6523  ‘cfv 6525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fo 6531  df-fv 6533

This theorem is referenced by:  basresprsfo  49608  basrestermcfo  50204