Theorem qusex 13190

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 2783	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
2	1	adantr 276	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → 𝑅 ∈ V)
3		elex 2783	. . . 4 ⊢ ( ∼ ∈ 𝑊 → ∼ ∈ V)
4	3	adantl 277	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → ∼ ∈ V)
5		basfn 12923	. . . . . 6 ⊢ Base Fn V
6		funfvex 5595	. . . . . . 7 ⊢ ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
7	6	funfni 5377	. . . . . 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 5813	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → (𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) ∈ V)
10		simpl 109	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → 𝑅 ∈ 𝑉)
11		imasex 13170	. . . 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 5578	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
14	13	mpteq1d 4130	. . . . 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 5964	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟) = ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥]𝑒) “s 𝑅))
17		eceq2 6659	. . . . . 6 ⊢ (𝑒 = ∼ → [𝑥]𝑒 = [𝑥] ∼ )
18	17	mpteq2dv 4136	. . . . 5 ⊢ (𝑒 = ∼ → (𝑥 ∈ (Base‘𝑅) ↦ [𝑥]𝑒) = (𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ))
19	18	oveq1d 5961	. . . 4 ⊢ (𝑒 = ∼ → ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥]𝑒) “s 𝑅) = ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) “s 𝑅))
20		df-qus 13168	. . . 4 ⊢ /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟))
21	16, 19, 20	ovmpog 6082	. . 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 1250	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → (𝑅 /s ∼ ) = ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ∼ ) “s 𝑅))
23	22, 12	eqeltrd 2282	1 ⊢ ((𝑅 ∈ 𝑉 ∧ ∼ ∈ 𝑊) → (𝑅 /s ∼ ) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2176  Vcvv 2772   ↦ cmpt 4106   Fn wfn 5267  ‘cfv 5272  (class class class)co 5946  [cec 6620  Basecbs 12865   “_s cimas 13164   /_s cqus 13165

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1re 8021  ax-addrcl 8024

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-tp 3641  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-ov 5949  df-oprab 5950  df-mpo 5951  df-ec 6624  df-inn 9039  df-2 9097  df-3 9098  df-ndx 12868  df-slot 12869  df-base 12871  df-plusg 12955  df-mulr 12956  df-iimas 13167  df-qus 13168

This theorem is referenced by:  znval  14431  znle  14432  znbaslemnn  14434