Theorem slotresfo 49252

Description: The condition of a structure component extractor restricted to a class being a surjection. This combined with fonex 49220 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 3960	. . . 4 ⊢ 𝐴 ⊆ V
3		fnssres 6623	. . . 4 ⊢ ((𝐸 Fn V ∧ 𝐴 ⊆ V) → (𝐸 ↾ 𝐴) Fn 𝐴)
4	1, 2, 3	mp2an 693	. . 3 ⊢ (𝐸 ↾ 𝐴) Fn 𝐴
5		fvres 6861	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 → ((𝐸 ↾ 𝐴)‘𝑘) = (𝐸‘𝑘))
6		slotresfo.v	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 → (𝐸‘𝑘) ∈ 𝑉)
7	5, 6	eqeltrd 2837	. . . . 5 ⊢ (𝑘 ∈ 𝐴 → ((𝐸 ↾ 𝐴)‘𝑘) ∈ 𝑉)
8	7	rgen 3054	. . . 4 ⊢ ∀𝑘 ∈ 𝐴 ((𝐸 ↾ 𝐴)‘𝑘) ∈ 𝑉
9		fnfvrnss 7075	. . . 4 ⊢ (((𝐸 ↾ 𝐴) Fn 𝐴 ∧ ∀𝑘 ∈ 𝐴 ((𝐸 ↾ 𝐴)‘𝑘) ∈ 𝑉) → ran (𝐸 ↾ 𝐴) ⊆ 𝑉)
10	4, 8, 9	mp2an 693	. . 3 ⊢ ran (𝐸 ↾ 𝐴) ⊆ 𝑉
11		df-f 6504	. . 3 ⊢ ((𝐸 ↾ 𝐴):𝐴⟶𝑉 ↔ ((𝐸 ↾ 𝐴) Fn 𝐴 ∧ ran (𝐸 ↾ 𝐴) ⊆ 𝑉))
12	4, 10, 11	mpbir2an 712	. 2 ⊢ (𝐸 ↾ 𝐴):𝐴⟶𝑉
13		fveq2 6842	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝐸‘𝑘) = (𝐸‘𝐾))
14	13	eqeq2d 2748	. . . . 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 3581	. . . 4 ⊢ (𝑏 ∈ 𝑉 → ∃𝑘 ∈ 𝐴 𝑏 = (𝐸‘𝑘))
18	5	eqeq2d 2748	. . . . 5 ⊢ (𝑘 ∈ 𝐴 → (𝑏 = ((𝐸 ↾ 𝐴)‘𝑘) ↔ 𝑏 = (𝐸‘𝑘)))
19	18	rexbiia 3083	. . . 4 ⊢ (∃𝑘 ∈ 𝐴 𝑏 = ((𝐸 ↾ 𝐴)‘𝑘) ↔ ∃𝑘 ∈ 𝐴 𝑏 = (𝐸‘𝑘))
20	17, 19	sylibr 234	. . 3 ⊢ (𝑏 ∈ 𝑉 → ∃𝑘 ∈ 𝐴 𝑏 = ((𝐸 ↾ 𝐴)‘𝑘))
21	20	rgen 3054	. 2 ⊢ ∀𝑏 ∈ 𝑉 ∃𝑘 ∈ 𝐴 𝑏 = ((𝐸 ↾ 𝐴)‘𝑘)
22		dffo3 7056	. 2 ⊢ ((𝐸 ↾ 𝐴):𝐴–onto→𝑉 ↔ ((𝐸 ↾ 𝐴):𝐴⟶𝑉 ∧ ∀𝑏 ∈ 𝑉 ∃𝑘 ∈ 𝐴 𝑏 = ((𝐸 ↾ 𝐴)‘𝑘)))
23	12, 21, 22	mpbir2an 712	1 ⊢ (𝐸 ↾ 𝐴):𝐴–onto→𝑉

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3442   ⊆ wss 3903  ran crn 5633   ↾ cres 5634   Fn wfn 6495  ⟶wf 6496  –onto→wfo 6498  ‘cfv 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508

This theorem is referenced by:  basresprsfo  49332  basrestermcfo  49928