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Mirrors > Home > ILE Home > Th. List > strslfv3 | GIF version |
Description: Variant on strslfv 12375 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 12375 | . . . 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 5484 | . . 3 ⊢ (𝜑 → (𝐸‘𝑈) = (𝐸‘𝑆)) |
10 | 7, 9 | eqtr4d 2200 | . 2 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑈)) |
11 | 1, 10 | eqtr4id 2216 | 1 ⊢ (𝜑 → 𝐴 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 ⊆ wss 3111 {csn 3570 〈cop 3573 class class class wbr 3976 ‘cfv 5182 ℕcn 8848 Struct cstr 12327 ndxcnx 12328 Slot cslot 12330 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-iota 5147 df-fun 5184 df-fv 5190 df-struct 12333 df-slot 12335 |
This theorem is referenced by: (None) |
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