Theorem slotresfo 49951

Description: The condition of a structure component extractor restricted to a class being a surjection. This combined with fonex 49921 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 3955	. . . 4 ⊢ 𝐴 ⊆ V
3		fnssres 6654	. . . 4 ⊢ ((𝐸 Fn V ∧ 𝐴 ⊆ V) → (𝐸 ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 705	. . 3 ⊢ (𝐸 ↾ 𝐴) Fn 𝐴
5		fvres 6896	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 → ((𝐸 ↾ 𝐴)‘𝑘) = (𝐸‘𝑘))
6		slotresfo.v	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 → (𝐸‘𝑘) ∈ 𝑉)
7	5, 6	eqeltrd 2861	. . . . 5 ⊢ (𝑘 ∈ 𝐴 → ((𝐸 ↾ 𝐴)‘𝑘) ∈ 𝑉)
8	7	rgen 3079	. . . 4 ⊢ ∀𝑘 ∈ 𝐴 ((𝐸 ↾ 𝐴)‘𝑘) ∈ 𝑉
9		fnfvrnss 7113	. . . 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 6535	. . 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 6877	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝐸‘𝑘) = (𝐸‘𝐾))
14	13	eqeq2d 2772	. . . . 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 3580	. . . 4 ⊢ (𝑏 ∈ 𝑉 → ∃𝑘 ∈ 𝐴 𝑏 = (𝐸‘𝑘))
18	5	eqeq2d 2772	. . . . 5 ⊢ (𝑘 ∈ 𝐴 → (𝑏 = ((𝐸 ↾ 𝐴)‘𝑘) ↔ 𝑏 = (𝐸‘𝑘)))
19	18	rexbiia 3108	. . . 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 3079	. 2 ⊢ ∀𝑏 ∈ 𝑉 ∃𝑘 ∈ 𝐴 𝑏 = ((𝐸 ↾ 𝐴)‘𝑘)
22		dffo3 7094	. 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 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  Vcvv 3451   ⊆ wss 3899  ran crn 5652   ↾ cres 5653   Fn wfn 6526  ⟶wf 6527  –onto→wfo 6529  ‘cfv 6531

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-fo 6537  df-fv 6539

This theorem is used by:  basresprsfo  50031  basrestermcfo  50627