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Mirrors > Home > ILE Home > Th. List > strslfv3 | GIF version |
Description: Variant on strslfv 12497 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 12497 | . . . 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 5516 | . . 3 ⊢ (𝜑 → (𝐸‘𝑈) = (𝐸‘𝑆)) |
10 | 7, 9 | eqtr4d 2213 | . 2 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑈)) |
11 | 1, 10 | eqtr4id 2229 | 1 ⊢ (𝜑 → 𝐴 = 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ⊆ wss 3129 {csn 3592 〈cop 3595 class class class wbr 4001 ‘cfv 5213 ℕcn 8913 Struct cstr 12448 ndxcnx 12449 Slot cslot 12451 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4119 ax-pow 4172 ax-pr 4207 ax-un 4431 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3809 df-br 4002 df-opab 4063 df-mpt 4064 df-id 4291 df-xp 4630 df-rel 4631 df-cnv 4632 df-co 4633 df-dm 4634 df-rn 4635 df-res 4636 df-iota 5175 df-fun 5215 df-fv 5221 df-struct 12454 df-slot 12456 |
This theorem is referenced by: (None) |
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