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Mirrors > Home > ILE Home > Th. List > strslfv2d | GIF version |
Description: Deduction version of strslfv 11846. (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 11822 | . 2 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘(𝐸‘ndx))) |
7 | cnvcnv2 4950 | . . . 4 ⊢ ◡◡𝑆 = (𝑆 ↾ V) | |
8 | 7 | fveq1i 5376 | . . 3 ⊢ (◡◡𝑆‘(𝐸‘ndx)) = ((𝑆 ↾ V)‘(𝐸‘ndx)) |
9 | 5 | elexd 2670 | . . . 4 ⊢ (𝜑 → (𝐸‘ndx) ∈ V) |
10 | fvres 5399 | . . . 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 2159 | . 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 2670 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ V) |
17 | 9, 16 | opelxpd 4532 | . . . . 5 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ (V × V)) |
18 | 14, 17 | elind 3227 | . . . 4 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ (𝑆 ∩ (V × V))) |
19 | cnvcnv 4949 | . . . 4 ⊢ ◡◡𝑆 = (𝑆 ∩ (V × V)) | |
20 | 18, 19 | syl6eleqr 2208 | . . 3 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ ◡◡𝑆) |
21 | funopfv 5415 | . . 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 2154 | 1 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1314 ∈ wcel 1463 Vcvv 2657 ∩ cin 3036 〈cop 3496 × cxp 4497 ◡ccnv 4498 ↾ cres 4501 Fun wfun 5075 ‘cfv 5081 ℕcn 8630 ndxcnx 11799 Slot cslot 11801 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-v 2659 df-sbc 2879 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-mpt 3951 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-iota 5046 df-fun 5083 df-fv 5089 df-slot 11806 |
This theorem is referenced by: strslfv2 11845 strslfv 11846 opelstrsl 11898 |
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