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Mirrors > Home > ILE Home > Th. List > strnfvnd | GIF version |
Description: Deduction version of strnfvn 12358. (Contributed by Mario Carneiro, 15-Nov-2014.) (Revised by Jim Kingdon, 19-Jan-2023.) |
Ref | Expression |
---|---|
strnfvnd.c | ⊢ 𝐸 = Slot 𝑁 |
strnfvnd.f | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
strnfvnd.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
Ref | Expression |
---|---|
strnfvnd | ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strnfvnd.f | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
2 | 1 | elexd 2734 | . 2 ⊢ (𝜑 → 𝑆 ∈ V) |
3 | strnfvnd.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
4 | fvexg 5499 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑆‘𝑁) ∈ V) | |
5 | 1, 3, 4 | syl2anc 409 | . 2 ⊢ (𝜑 → (𝑆‘𝑁) ∈ V) |
6 | fveq1 5479 | . . 3 ⊢ (𝑥 = 𝑆 → (𝑥‘𝑁) = (𝑆‘𝑁)) | |
7 | strnfvnd.c | . . . 4 ⊢ 𝐸 = Slot 𝑁 | |
8 | df-slot 12341 | . . . 4 ⊢ Slot 𝑁 = (𝑥 ∈ V ↦ (𝑥‘𝑁)) | |
9 | 7, 8 | eqtri 2185 | . . 3 ⊢ 𝐸 = (𝑥 ∈ V ↦ (𝑥‘𝑁)) |
10 | 6, 9 | fvmptg 5556 | . 2 ⊢ ((𝑆 ∈ V ∧ (𝑆‘𝑁) ∈ V) → (𝐸‘𝑆) = (𝑆‘𝑁)) |
11 | 2, 5, 10 | syl2anc 409 | 1 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 Vcvv 2721 ↦ cmpt 4037 ‘cfv 5182 ℕcn 8848 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: strnfvn 12358 strfvssn 12359 strndxid 12365 strsetsid 12370 strslfvd 12378 strslfv2d 12379 setsslid 12387 setsslnid 12388 ressid 12398 |
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