Theorem qusex 13646

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 2833	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2	1	adantr 276	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → 𝑅 ∈ V)
3		elex 2833	. . . 4 ⊢ ( ∼ ∈ 𝑊 → ∼ ∈ V)
4	3	adantl 277	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → ∼ ∈ V)
5		basfn 13411	. . . . . 6 ⊢ Base Fn V
6		funfvex 5712	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
7	6	funfni 5483	. . . . . 6 ⊢ ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
8	5, 2, 7	sylancr 418	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → (Base‘𝑅) ∈ V)
9	8	mptexd 5944	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → (𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) ∈ V)
10		simpl 109	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → 𝑅 ∈ 𝑉)
11		imasex 13626	. . . 4 ⊢ (((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) ∈ V ∧ 𝑅 ∈ 𝑉) → ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) “s 𝑅) ∈ V)
12	9, 10, 11	syl2anc 415	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) “s 𝑅) ∈ V)
13		fveq2 5695	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
14	13	mpteq1d 4216	. . . . 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 6103	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟) = ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥]𝑒) “s 𝑅))
17		eceq2 6844	. . . . . 6 ⊢ (𝑒 = ∼ → [𝑥]𝑒 = [𝑥] ∼ )
18	17	mpteq2dv 4222	. . . . 5 ⊢ (𝑒 = ∼ → (𝑥 ∈ (Base‘𝑅) ↦ [𝑥]𝑒) = (𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ))
19	18	oveq1d 6100	. . . 4 ⊢ (𝑒 = ∼ → ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥]𝑒) “s 𝑅) = ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) “s 𝑅))
20		df-qus 13625	. . . 4 ⊢ /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟))
21	16, 19, 20	ovmpog 6223	. . 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 1278	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → (𝑅 /s ∼ ) = ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) “s 𝑅))
23	22, 12	eqeltrd 2315	1 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → (𝑅 /s ∼ ) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ↦ cmpt 4192   Fn wfn 5372  ‘cfv 5377  (class class class)co 6085  [cec 6805  Basecbs 13352   “_s cimas 13622   /_s cqus 13623

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-ec 6809  df-inn 9305  df-2 9363  df-3 9364  df-ndx 13355  df-slot 13356  df-base 13358  df-plusg 13444  df-mulr 13445  df-iimas 13624  df-qus 13625

This theorem is used by:  znval  14971  znle  14972  znbaslemnn  14974