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Mirrors > Home > ILE Home > Th. List > strslfv3 | GIF version |
Description: Variant on strslfv 11930 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 11930 | . . . 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 5393 | . . 3 ⊢ (𝜑 → (𝐸‘𝑈) = (𝐸‘𝑆)) |
9 | 6, 8 | eqtr4d 2153 | . 2 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑈)) |
10 | strfv3.a | . 2 ⊢ 𝐴 = (𝐸‘𝑈) | |
11 | 9, 10 | syl6reqr 2169 | 1 ⊢ (𝜑 → 𝐴 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 ⊆ wss 3041 {csn 3497 〈cop 3500 class class class wbr 3899 ‘cfv 5093 ℕcn 8688 Struct cstr 11882 ndxcnx 11883 Slot cslot 11885 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-iota 5058 df-fun 5095 df-fv 5101 df-struct 11888 df-slot 11890 |
This theorem is referenced by: (None) |
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