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Theorem strslfv3 12498
Description: Variant on strslfv 12497 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.) (Revised by Jim Kingdon, 30-Jan-2023.)
Hypotheses
Ref Expression
strfv3.u (𝜑𝑈 = 𝑆)
strfv3.s 𝑆 Struct 𝑋
strslfv3.e (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
strfv3.n {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆
strfv3.c (𝜑𝐶𝑉)
strfv3.a 𝐴 = (𝐸𝑈)
Assertion
Ref Expression
strslfv3 (𝜑𝐴 = 𝐶)

Proof of Theorem strslfv3
StepHypRef Expression
1 strfv3.a . 2 𝐴 = (𝐸𝑈)
2 strfv3.c . . . 4 (𝜑𝐶𝑉)
3 strfv3.s . . . . 5 𝑆 Struct 𝑋
4 strslfv3.e . . . . 5 (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)
5 strfv3.n . . . . 5 {⟨(𝐸‘ndx), 𝐶⟩} ⊆ 𝑆
63, 4, 5strslfv 12497 . . . 4 (𝐶𝑉𝐶 = (𝐸𝑆))
72, 6syl 14 . . 3 (𝜑𝐶 = (𝐸𝑆))
8 strfv3.u . . . 4 (𝜑𝑈 = 𝑆)
98fveq2d 5516 . . 3 (𝜑 → (𝐸𝑈) = (𝐸𝑆))
107, 9eqtr4d 2213 . 2 (𝜑𝐶 = (𝐸𝑈))
111, 10eqtr4id 2229 1 (𝜑𝐴 = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  wss 3129  {csn 3592  cop 3595   class class class wbr 4001  cfv 5213  cn 8913   Struct cstr 12448  ndxcnx 12449  Slot cslot 12451
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 614  ax-in2 615  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 4119  ax-pow 4172  ax-pr 4207  ax-un 4431
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  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-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-br 4002  df-opab 4063  df-mpt 4064  df-id 4291  df-xp 4630  df-rel 4631  df-cnv 4632  df-co 4633  df-dm 4634  df-rn 4635  df-res 4636  df-iota 5175  df-fun 5215  df-fv 5221  df-struct 12454  df-slot 12456
This theorem is referenced by: (None)
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