Theorem qusex 12968

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 2774	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2	1	adantr 276	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → 𝑅 ∈ V)
3		elex 2774	. . . 4 ⊢ ( ∼ ∈ 𝑊 → ∼ ∈ V)
4	3	adantl 277	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → ∼ ∈ V)
5		basfn 12736	. . . . . 6 ⊢ Base Fn V
6		funfvex 5575	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
7	6	funfni 5358	. . . . . 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 5789	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → (𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) ∈ V)
10		simpl 109	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → 𝑅 ∈ 𝑉)
11		imasex 12948	. . . 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 5558	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
14	13	mpteq1d 4118	. . . . 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 5940	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟) = ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥]𝑒) “s 𝑅))
17		eceq2 6629	. . . . . 6 ⊢ (𝑒 = ∼ → [𝑥]𝑒 = [𝑥] ∼ )
18	17	mpteq2dv 4124	. . . . 5 ⊢ (𝑒 = ∼ → (𝑥 ∈ (Base‘𝑅) ↦ [𝑥]𝑒) = (𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ))
19	18	oveq1d 5937	. . . 4 ⊢ (𝑒 = ∼ → ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥]𝑒) “s 𝑅) = ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) “s 𝑅))
20		df-qus 12946	. . . 4 ⊢ /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟))
21	16, 19, 20	ovmpog 6057	. . 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 1249	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → (𝑅 /s ∼ ) = ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) “s 𝑅))
23	22, 12	eqeltrd 2273	1 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → (𝑅 /s ∼ ) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  Vcvv 2763   ↦ cmpt 4094   Fn wfn 5253  ‘cfv 5258  (class class class)co 5922  [cec 6590  Basecbs 12678   “_s cimas 12942   /_s cqus 12943

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-tp 3630  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-ec 6594  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-mulr 12769  df-iimas 12945  df-qus 12946

This theorem is referenced by:  znval  14192  znle  14193  znbaslemnn  14195