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Mirrors > Home > ILE Home > Th. List > strslssd | GIF version |
Description: Deduction version of strslss 12006. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by Jim Kingdon, 31-Jan-2023.) |
Ref | Expression |
---|---|
strslssd.e | ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) |
strssd.t | ⊢ (𝜑 → 𝑇 ∈ 𝑉) |
strssd.f | ⊢ (𝜑 → Fun 𝑇) |
strssd.s | ⊢ (𝜑 → 𝑆 ⊆ 𝑇) |
strssd.n | ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆) |
Ref | Expression |
---|---|
strslssd | ⊢ (𝜑 → (𝐸‘𝑇) = (𝐸‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strslssd.e | . . 3 ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) | |
2 | strssd.t | . . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑉) | |
3 | strssd.f | . . 3 ⊢ (𝜑 → Fun 𝑇) | |
4 | strssd.s | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑇) | |
5 | strssd.n | . . . 4 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆) | |
6 | 4, 5 | sseldd 3098 | . . 3 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑇) |
7 | 1, 2, 3, 6 | strslfvd 12000 | . 2 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑇)) |
8 | 2, 4 | ssexd 4068 | . . 3 ⊢ (𝜑 → 𝑆 ∈ V) |
9 | funss 5142 | . . . 4 ⊢ (𝑆 ⊆ 𝑇 → (Fun 𝑇 → Fun 𝑆)) | |
10 | 4, 3, 9 | sylc 62 | . . 3 ⊢ (𝜑 → Fun 𝑆) |
11 | 1, 8, 10, 5 | strslfvd 12000 | . 2 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆)) |
12 | 7, 11 | eqtr3d 2174 | 1 ⊢ (𝜑 → (𝐸‘𝑇) = (𝐸‘𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ⊆ wss 3071 〈cop 3530 Fun wfun 5117 ‘cfv 5123 ℕcn 8720 ndxcnx 11956 Slot cslot 11958 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fv 5131 df-slot 11963 |
This theorem is referenced by: strslss 12006 |
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