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Theorem strslfv3 12461
Description: Variant on strslfv 12460 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 12460 . . . 4 (𝐶𝑉𝐶 = (𝐸𝑆))
72, 6syl 14 . . 3 (𝜑𝐶 = (𝐸𝑆))
8 strfv3.u . . . 4 (𝜑𝑈 = 𝑆)
98fveq2d 5500 . . 3 (𝜑 → (𝐸𝑈) = (𝐸𝑆))
107, 9eqtr4d 2206 . 2 (𝜑𝐶 = (𝐸𝑈))
111, 10eqtr4id 2222 1 (𝜑𝐴 = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  wcel 2141  wss 3121  {csn 3583  cop 3586   class class class wbr 3989  cfv 5198  cn 8878   Struct cstr 12412  ndxcnx 12413  Slot cslot 12415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fv 5206  df-struct 12418  df-slot 12420
This theorem is referenced by: (None)
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