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Mirrors > Home > ILE Home > Th. List > strslfv2d | GIF version |
Description: Deduction version of strslfv 12521. (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 111 | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) |
3 | strfv2d.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
4 | 1 | simpri 113 | . . . 4 ⊢ (𝐸‘ndx) ∈ ℕ |
5 | 4 | a1i 9 | . . 3 ⊢ (𝜑 → (𝐸‘ndx) ∈ ℕ) |
6 | 2, 3, 5 | strnfvnd 12496 | . 2 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘(𝐸‘ndx))) |
7 | cnvcnv2 5094 | . . . 4 ⊢ ◡◡𝑆 = (𝑆 ↾ V) | |
8 | 7 | fveq1i 5528 | . . 3 ⊢ (◡◡𝑆‘(𝐸‘ndx)) = ((𝑆 ↾ V)‘(𝐸‘ndx)) |
9 | 5 | elexd 2762 | . . . 4 ⊢ (𝜑 → (𝐸‘ndx) ∈ V) |
10 | fvres 5551 | . . . 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 | eqtrid 2232 | . 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 2762 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ V) |
17 | 9, 16 | opelxpd 4671 | . . . . 5 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ (V × V)) |
18 | 14, 17 | elind 3332 | . . . 4 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ (𝑆 ∩ (V × V))) |
19 | cnvcnv 5093 | . . . 4 ⊢ ◡◡𝑆 = (𝑆 ∩ (V × V)) | |
20 | 18, 19 | eleqtrrdi 2281 | . . 3 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ ◡◡𝑆) |
21 | funopfv 5568 | . . 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 2227 | 1 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1363 ∈ wcel 2158 Vcvv 2749 ∩ cin 3140 〈cop 3607 × cxp 4636 ◡ccnv 4637 ↾ cres 4640 Fun wfun 5222 ‘cfv 5228 ℕcn 8933 ndxcnx 12473 Slot cslot 12475 |
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-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-sbc 2975 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-iota 5190 df-fun 5230 df-fv 5236 df-slot 12480 |
This theorem is referenced by: strslfv2 12520 strslfv 12521 opelstrsl 12588 |
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