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Theorem qusex 12795
Description: Existence of a quotient structure. (Contributed by Jim Kingdon, 25-Apr-2025.)
Assertion
Ref Expression
qusex ((𝑅𝑉𝑊) → (𝑅 /s ) ∈ V)

Proof of Theorem qusex
Dummy variables 𝑒 𝑟 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2763 . . . 4 (𝑅𝑉𝑅 ∈ V)
21adantr 276 . . 3 ((𝑅𝑉𝑊) → 𝑅 ∈ V)
3 elex 2763 . . . 4 ( 𝑊 ∈ V)
43adantl 277 . . 3 ((𝑅𝑉𝑊) → ∈ V)
5 basfn 12565 . . . . . 6 Base Fn V
6 funfvex 5548 . . . . . . 7 ((Fun Base ∧ 𝑅 ∈ dom Base) → (Base‘𝑅) ∈ V)
76funfni 5332 . . . . . 6 ((Base Fn V ∧ 𝑅 ∈ V) → (Base‘𝑅) ∈ V)
85, 2, 7sylancr 414 . . . . 5 ((𝑅𝑉𝑊) → (Base‘𝑅) ∈ V)
98mptexd 5760 . . . 4 ((𝑅𝑉𝑊) → (𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ) ∈ V)
10 simpl 109 . . . 4 ((𝑅𝑉𝑊) → 𝑅𝑉)
11 imasex 12775 . . . 4 (((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ) ∈ V ∧ 𝑅𝑉) → ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ) “s 𝑅) ∈ V)
129, 10, 11syl2anc 411 . . 3 ((𝑅𝑉𝑊) → ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ) “s 𝑅) ∈ V)
13 fveq2 5531 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
1413mpteq1d 4103 . . . . 5 (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) = (𝑥 ∈ (Base‘𝑅) ↦ [𝑥]𝑒))
15 id 19 . . . . 5 (𝑟 = 𝑅𝑟 = 𝑅)
1614, 15oveq12d 5910 . . . 4 (𝑟 = 𝑅 → ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟) = ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥]𝑒) “s 𝑅))
17 eceq2 6591 . . . . . 6 (𝑒 = → [𝑥]𝑒 = [𝑥] )
1817mpteq2dv 4109 . . . . 5 (𝑒 = → (𝑥 ∈ (Base‘𝑅) ↦ [𝑥]𝑒) = (𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ))
1918oveq1d 5907 . . . 4 (𝑒 = → ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥]𝑒) “s 𝑅) = ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ) “s 𝑅))
20 df-qus 12773 . . . 4 /s = (𝑟 ∈ V, 𝑒 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟) ↦ [𝑥]𝑒) “s 𝑟))
2116, 19, 20ovmpog 6027 . . 3 ((𝑅 ∈ V ∧ ∈ V ∧ ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ) “s 𝑅) ∈ V) → (𝑅 /s ) = ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ) “s 𝑅))
222, 4, 12, 21syl3anc 1249 . 2 ((𝑅𝑉𝑊) → (𝑅 /s ) = ((𝑥 ∈ (Base‘𝑅) ↦ [𝑥] ) “s 𝑅))
2322, 12eqeltrd 2266 1 ((𝑅𝑉𝑊) → (𝑅 /s ) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  Vcvv 2752  cmpt 4079   Fn wfn 5227  cfv 5232  (class class class)co 5892  [cec 6552  Basecbs 12507  s cimas 12769   /s cqus 12770
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7927  ax-resscn 7928  ax-1re 7930  ax-addrcl 7933
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-tp 3615  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-ov 5895  df-oprab 5896  df-mpo 5897  df-ec 6556  df-inn 8945  df-2 9003  df-3 9004  df-ndx 12510  df-slot 12511  df-base 12513  df-plusg 12595  df-mulr 12596  df-iimas 12772  df-qus 12773
This theorem is referenced by:  znval  13925  znle  13926  znbaslemnn  13928
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