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Theorem ressval3d 12545
Description: Value of structure restriction, deduction version. (Contributed by AV, 14-Mar-2020.) (Revised by Jim Kingdon, 17-Jan-2025.)
Hypotheses
Ref Expression
ressval3d.r 𝑅 = (𝑆 β†Ύs 𝐴)
ressval3d.b 𝐡 = (Baseβ€˜π‘†)
ressval3d.e 𝐸 = (Baseβ€˜ndx)
ressval3d.s (πœ‘ β†’ 𝑆 ∈ 𝑉)
ressval3d.f (πœ‘ β†’ Fun 𝑆)
ressval3d.d (πœ‘ β†’ 𝐸 ∈ dom 𝑆)
ressval3d.u (πœ‘ β†’ 𝐴 βŠ† 𝐡)
Assertion
Ref Expression
ressval3d (πœ‘ β†’ 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))

Proof of Theorem ressval3d
StepHypRef Expression
1 ressval3d.r . . 3 𝑅 = (𝑆 β†Ύs 𝐴)
2 ressval3d.s . . . 4 (πœ‘ β†’ 𝑆 ∈ 𝑉)
3 ressval3d.b . . . . . 6 𝐡 = (Baseβ€˜π‘†)
4 basfn 12533 . . . . . . 7 Base Fn V
52elexd 2762 . . . . . . 7 (πœ‘ β†’ 𝑆 ∈ V)
6 funfvex 5544 . . . . . . . 8 ((Fun Base ∧ 𝑆 ∈ dom Base) β†’ (Baseβ€˜π‘†) ∈ V)
76funfni 5328 . . . . . . 7 ((Base Fn V ∧ 𝑆 ∈ V) β†’ (Baseβ€˜π‘†) ∈ V)
84, 5, 7sylancr 414 . . . . . 6 (πœ‘ β†’ (Baseβ€˜π‘†) ∈ V)
93, 8eqeltrid 2274 . . . . 5 (πœ‘ β†’ 𝐡 ∈ V)
10 ressval3d.u . . . . 5 (πœ‘ β†’ 𝐴 βŠ† 𝐡)
119, 10ssexd 4155 . . . 4 (πœ‘ β†’ 𝐴 ∈ V)
12 ressvalsets 12537 . . . 4 ((𝑆 ∈ 𝑉 ∧ 𝐴 ∈ V) β†’ (𝑆 β†Ύs 𝐴) = (𝑆 sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘†))⟩))
132, 11, 12syl2anc 411 . . 3 (πœ‘ β†’ (𝑆 β†Ύs 𝐴) = (𝑆 sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘†))⟩))
141, 13eqtrid 2232 . 2 (πœ‘ β†’ 𝑅 = (𝑆 sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘†))⟩))
15 ressval3d.e . . . . 5 𝐸 = (Baseβ€˜ndx)
1615a1i 9 . . . 4 (πœ‘ β†’ 𝐸 = (Baseβ€˜ndx))
17 df-ss 3154 . . . . . 6 (𝐴 βŠ† 𝐡 ↔ (𝐴 ∩ 𝐡) = 𝐴)
1810, 17sylib 122 . . . . 5 (πœ‘ β†’ (𝐴 ∩ 𝐡) = 𝐴)
193ineq2i 3345 . . . . 5 (𝐴 ∩ 𝐡) = (𝐴 ∩ (Baseβ€˜π‘†))
2018, 19eqtr3di 2235 . . . 4 (πœ‘ β†’ 𝐴 = (𝐴 ∩ (Baseβ€˜π‘†)))
2116, 20opeq12d 3798 . . 3 (πœ‘ β†’ ⟨𝐸, 𝐴⟩ = ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘†))⟩)
2221oveq2d 5904 . 2 (πœ‘ β†’ (𝑆 sSet ⟨𝐸, 𝐴⟩) = (𝑆 sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ (Baseβ€˜π‘†))⟩))
2314, 22eqtr4d 2223 1 (πœ‘ β†’ 𝑅 = (𝑆 sSet ⟨𝐸, 𝐴⟩))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   = wceq 1363   ∈ wcel 2158  Vcvv 2749   ∩ cin 3140   βŠ† wss 3141  βŸ¨cop 3607  dom cdm 4638  Fun wfun 5222   Fn wfn 5223  β€˜cfv 5228  (class class class)co 5888  ndxcnx 12472   sSet csts 12473  Basecbs 12475   β†Ύs cress 12476
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1re 7918  ax-addrcl 7921
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-inn 8933  df-ndx 12478  df-slot 12479  df-base 12481  df-sets 12482  df-iress 12483
This theorem is referenced by: (None)
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