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Mirrors > Home > ILE Home > Th. List > strslssd | GIF version |
Description: Deduction version of strslss 12384. (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 3138 | . . 3 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑇) |
7 | 1, 2, 3, 6 | strslfvd 12378 | . 2 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑇)) |
8 | 2, 4 | ssexd 4116 | . . 3 ⊢ (𝜑 → 𝑆 ∈ V) |
9 | funss 5201 | . . . 4 ⊢ (𝑆 ⊆ 𝑇 → (Fun 𝑇 → Fun 𝑆)) | |
10 | 4, 3, 9 | sylc 62 | . . 3 ⊢ (𝜑 → Fun 𝑆) |
11 | 1, 8, 10, 5 | strslfvd 12378 | . 2 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆)) |
12 | 7, 11 | eqtr3d 2199 | 1 ⊢ (𝜑 → (𝐸‘𝑇) = (𝐸‘𝑆)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 Vcvv 2721 ⊆ wss 3111 〈cop 3573 Fun wfun 5176 ‘cfv 5182 ℕcn 8848 ndxcnx 12334 Slot cslot 12336 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-iota 5147 df-fun 5184 df-fv 5190 df-slot 12341 |
This theorem is referenced by: strslss 12384 |
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