Theorem qusex 13538

Description: Existence of a quotient structure. (Contributed by Jim Kingdon, 25-Apr-2025.)

Assertion

Ref	Expression
qusex	⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → (𝑅 /s ∼ ) ∈ V)

Proof of Theorem qusex

Dummy variables 𝑒 𝑟 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2825	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2	1	adantr 276	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → 𝑅 ∈ V)
3		elex 2825	. . . 4 ⊢ ( ∼ ∈ 𝑊 → ∼ ∈ V)
4	3	adantl 277	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → ∼ ∈ V)
5		basfn 13271	. . . . . 6 ⊢ Base Fn V
6		funfvex 5687	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
7	6	funfni 5458	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
8	5, 2, 7	sylancr 414	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → (Base‘𝑅) ∈ V)
9	8	mptexd 5913	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → (𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) ∈ V)
10		simpl 109	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → 𝑅 ∈ 𝑉)
11		imasex 13518	. . . 4 ⊢ (((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) ∈ V ∧ 𝑅 ∈ 𝑉) → ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) “s 𝑅) ∈ V)
12	9, 10, 11	syl2anc 411	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) “s 𝑅) ∈ V)
13		fveq2 5670	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
14	13	mpteq1d 4195	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = (𝑥 ∈ (Base‘𝑅) ↦ [𝑥]𝑒))
15		id 19	. . . . 5 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅)
16	14, 15	oveq12d 6068	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟) = ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥]𝑒) “s 𝑅))
17		eceq2 6804	. . . . . 6 ⊢ (𝑒 = ∼ → [𝑥]𝑒 = [𝑥] ∼ )
18	17	mpteq2dv 4201	. . . . 5 ⊢ (𝑒 = ∼ → (𝑥 ∈ (Base‘𝑅) ↦ [𝑥]𝑒) = (𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ))
19	18	oveq1d 6065	. . . 4 ⊢ (𝑒 = ∼ → ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥]𝑒) “s 𝑅) = ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) “s 𝑅))
20		df-qus 13516	. . . 4 ⊢ /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟))
21	16, 19, 20	ovmpog 6188	. . 3 ⊢ ((𝑅 ∈ V ∧ ∼ ∈ V ∧ ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) “s 𝑅) ∈ V) → (𝑅 /s ∼ ) = ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) “s 𝑅))
22	2, 4, 12, 21	syl3anc 1274	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → (𝑅 /s ∼ ) = ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) “s 𝑅))
23	22, 12	eqeltrd 2309	1 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → (𝑅 /s ∼ ) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203  Vcvv 2813   ↦ cmpt 4171   Fn wfn 5347  ‘cfv 5352  (class class class)co 6050  [cec 6765  Basecbs 13212   “_s cimas 13512   /_s cqus 13513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-tp 3697  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-ec 6769  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-plusg 13303  df-mulr 13304  df-iimas 13515  df-qus 13516

This theorem is referenced by:  znval  14784  znle  14785  znbaslemnn  14787