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Mirrors > Home > ILE Home > Th. List > strslfv3 | GIF version |
Description: Variant on strslfv 12663 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.a | . 2 ⊢ 𝐴 = (𝐸‘𝑈) | |
2 | strfv3.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
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 | |
6 | 3, 4, 5 | strslfv 12663 | . . . 4 ⊢ (𝐶 ∈ 𝑉 → 𝐶 = (𝐸‘𝑆)) |
7 | 2, 6 | syl 14 | . . 3 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆)) |
8 | strfv3.u | . . . 4 ⊢ (𝜑 → 𝑈 = 𝑆) | |
9 | 8 | fveq2d 5558 | . . 3 ⊢ (𝜑 → (𝐸‘𝑈) = (𝐸‘𝑆)) |
10 | 7, 9 | eqtr4d 2229 | . 2 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑈)) |
11 | 1, 10 | eqtr4id 2245 | 1 ⊢ (𝜑 → 𝐴 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2164 ⊆ wss 3153 {csn 3618 〈cop 3621 class class class wbr 4029 ‘cfv 5254 ℕcn 8982 Struct cstr 12614 ndxcnx 12615 Slot cslot 12617 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-sbc 2986 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-iota 5215 df-fun 5256 df-fv 5262 df-struct 12620 df-slot 12622 |
This theorem is referenced by: (None) |
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