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Theorem strndxid 12460
Description: The value of a structure component extractor is the value of the corresponding slot of the structure. (Contributed by AV, 13-Mar-2020.)
Hypotheses
Ref Expression
strndxid.s (𝜑𝑆𝑉)
strndxid.e 𝐸 = Slot 𝑁
strndxid.n 𝑁 ∈ ℕ
Assertion
Ref Expression
strndxid (𝜑 → (𝑆‘(𝐸‘ndx)) = (𝐸𝑆))

Proof of Theorem strndxid
StepHypRef Expression
1 strndxid.e . . . 4 𝐸 = Slot 𝑁
2 strndxid.n . . . 4 𝑁 ∈ ℕ
31, 2ndxid 12456 . . 3 𝐸 = Slot (𝐸‘ndx)
4 strndxid.s . . 3 (𝜑𝑆𝑉)
51, 2ndxarg 12455 . . . . 5 (𝐸‘ndx) = 𝑁
65, 2eqeltri 2250 . . . 4 (𝐸‘ndx) ∈ ℕ
76a1i 9 . . 3 (𝜑 → (𝐸‘ndx) ∈ ℕ)
83, 4, 7strnfvnd 12452 . 2 (𝜑 → (𝐸𝑆) = (𝑆‘(𝐸‘ndx)))
98eqcomd 2183 1 (𝜑 → (𝑆‘(𝐸‘ndx)) = (𝐸𝑆))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  cfv 5211  cn 8895  ndxcnx 12429  Slot cslot 12431
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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-iota 5173  df-fun 5213  df-fv 5219  df-inn 8896  df-ndx 12435  df-slot 12436
This theorem is referenced by: (None)
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