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Mirrors > Home > ILE Home > Th. List > strslfv2d | GIF version |
Description: Deduction version of strslfv 11992. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 30-Jan-2023.) |
Ref | Expression |
---|---|
strslfv2d.e | ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) |
strfv2d.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
strfv2d.f | ⊢ (𝜑 → Fun ◡◡𝑆) |
strfv2d.n | ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆) |
strfv2d.c | ⊢ (𝜑 → 𝐶 ∈ 𝑊) |
Ref | Expression |
---|---|
strslfv2d | ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strslfv2d.e | . . . 4 ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) | |
2 | 1 | simpli 110 | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) |
3 | strfv2d.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
4 | 1 | simpri 112 | . . . 4 ⊢ (𝐸‘ndx) ∈ ℕ |
5 | 4 | a1i 9 | . . 3 ⊢ (𝜑 → (𝐸‘ndx) ∈ ℕ) |
6 | 2, 3, 5 | strnfvnd 11968 | . 2 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘(𝐸‘ndx))) |
7 | cnvcnv2 4987 | . . . 4 ⊢ ◡◡𝑆 = (𝑆 ↾ V) | |
8 | 7 | fveq1i 5415 | . . 3 ⊢ (◡◡𝑆‘(𝐸‘ndx)) = ((𝑆 ↾ V)‘(𝐸‘ndx)) |
9 | 5 | elexd 2694 | . . . 4 ⊢ (𝜑 → (𝐸‘ndx) ∈ V) |
10 | fvres 5438 | . . . 4 ⊢ ((𝐸‘ndx) ∈ V → ((𝑆 ↾ V)‘(𝐸‘ndx)) = (𝑆‘(𝐸‘ndx))) | |
11 | 9, 10 | syl 14 | . . 3 ⊢ (𝜑 → ((𝑆 ↾ V)‘(𝐸‘ndx)) = (𝑆‘(𝐸‘ndx))) |
12 | 8, 11 | syl5eq 2182 | . 2 ⊢ (𝜑 → (◡◡𝑆‘(𝐸‘ndx)) = (𝑆‘(𝐸‘ndx))) |
13 | strfv2d.f | . . 3 ⊢ (𝜑 → Fun ◡◡𝑆) | |
14 | strfv2d.n | . . . . 5 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆) | |
15 | strfv2d.c | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑊) | |
16 | 15 | elexd 2694 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ V) |
17 | 9, 16 | opelxpd 4567 | . . . . 5 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ (V × V)) |
18 | 14, 17 | elind 3256 | . . . 4 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ (𝑆 ∩ (V × V))) |
19 | cnvcnv 4986 | . . . 4 ⊢ ◡◡𝑆 = (𝑆 ∩ (V × V)) | |
20 | 18, 19 | eleqtrrdi 2231 | . . 3 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ ◡◡𝑆) |
21 | funopfv 5454 | . . 3 ⊢ (Fun ◡◡𝑆 → (〈(𝐸‘ndx), 𝐶〉 ∈ ◡◡𝑆 → (◡◡𝑆‘(𝐸‘ndx)) = 𝐶)) | |
22 | 13, 20, 21 | sylc 62 | . 2 ⊢ (𝜑 → (◡◡𝑆‘(𝐸‘ndx)) = 𝐶) |
23 | 6, 12, 22 | 3eqtr2rd 2177 | 1 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 Vcvv 2681 ∩ cin 3065 〈cop 3525 × cxp 4532 ◡ccnv 4533 ↾ cres 4536 Fun wfun 5112 ‘cfv 5118 ℕcn 8713 ndxcnx 11945 Slot cslot 11947 |
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-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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-iota 5083 df-fun 5120 df-fv 5126 df-slot 11952 |
This theorem is referenced by: strslfv2 11991 strslfv 11992 opelstrsl 12044 |
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