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Mirrors > Home > ILE Home > Th. List > strslfv3 | GIF version |
Description: Variant on strslfv 12006 for large structures. (Contributed by Mario Carneiro, 10-Jan-2017.) (Revised by Jim Kingdon, 30-Jan-2023.) |
Ref | Expression |
---|---|
strfv3.u | ⊢ (𝜑 → 𝑈 = 𝑆) |
strfv3.s | ⊢ 𝑆 Struct 𝑋 |
strslfv3.e | ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) |
strfv3.n | ⊢ {〈(𝐸‘ndx), 𝐶〉} ⊆ 𝑆 |
strfv3.c | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
strfv3.a | ⊢ 𝐴 = (𝐸‘𝑈) |
Ref | Expression |
---|---|
strslfv3 | ⊢ (𝜑 → 𝐴 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strfv3.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
2 | strfv3.s | . . . . 5 ⊢ 𝑆 Struct 𝑋 | |
3 | strslfv3.e | . . . . 5 ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) | |
4 | strfv3.n | . . . . 5 ⊢ {〈(𝐸‘ndx), 𝐶〉} ⊆ 𝑆 | |
5 | 2, 3, 4 | strslfv 12006 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆)) |
6 | 1, 5 | syl 14 | . . 3 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆)) |
7 | strfv3.u | . . . 4 ⊢ (𝜑 → 𝑈 = 𝑆) | |
8 | 7 | fveq2d 5425 | . . 3 ⊢ (𝜑 → (𝐸‘𝑈) = (𝐸‘𝑆)) |
9 | 6, 8 | eqtr4d 2175 | . 2 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑈)) |
10 | strfv3.a | . 2 ⊢ 𝐴 = (𝐸‘𝑈) | |
11 | 9, 10 | syl6reqr 2191 | 1 ⊢ (𝜑 → 𝐴 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ⊆ wss 3071 {csn 3527 〈cop 3530 class class class wbr 3929 ‘cfv 5123 ℕcn 8723 Struct cstr 11958 ndxcnx 11959 Slot cslot 11961 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-iota 5088 df-fun 5125 df-fv 5131 df-struct 11964 df-slot 11966 |
This theorem is referenced by: (None) |
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