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Mirrors > Home > ILE Home > Th. List > strnfvnd | GIF version |
Description: Deduction version of strnfvn 11980. (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 2699 | . 2 ⊢ (𝜑 → 𝑆 ∈ V) |
3 | strnfvnd.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
4 | fvexg 5440 | . . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑆‘𝑁) ∈ V) | |
5 | 1, 3, 4 | syl2anc 408 | . 2 ⊢ (𝜑 → (𝑆‘𝑁) ∈ V) |
6 | fveq1 5420 | . . 3 ⊢ (𝑥 = 𝑆 → (𝑥‘𝑁) = (𝑆‘𝑁)) | |
7 | strnfvnd.c | . . . 4 ⊢ 𝐸 = Slot 𝑁 | |
8 | df-slot 11963 | . . . 4 ⊢ Slot 𝑁 = (𝑥 ∈ V ↦ (𝑥‘𝑁)) | |
9 | 7, 8 | eqtri 2160 | . . 3 ⊢ 𝐸 = (𝑥 ∈ V ↦ (𝑥‘𝑁)) |
10 | 6, 9 | fvmptg 5497 | . 2 ⊢ ((𝑆 ∈ V ∧ (𝑆‘𝑁) ∈ V) → (𝐸‘𝑆) = (𝑆‘𝑁)) |
11 | 2, 5, 10 | syl2anc 408 | 1 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ↦ cmpt 3989 ‘cfv 5123 ℕcn 8720 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: strnfvn 11980 strfvssn 11981 strndxid 11987 strsetsid 11992 strslfvd 12000 strslfv2d 12001 setsslid 12009 setsslnid 12010 ressid 12020 |
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