| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > strfvnd | Structured version Visualization version GIF version | ||
| Description: Deduction version of strfvn 17223. (Contributed by Mario Carneiro, 15-Nov-2014.) |
| Ref | Expression |
|---|---|
| strfvnd.c | ⊢ 𝐸 = Slot 𝑁 |
| strfvnd.f | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
| Ref | Expression |
|---|---|
| strfvnd | ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘𝑁)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | strfvnd.f | . 2 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
| 2 | elex 3501 | . 2 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) | |
| 3 | fveq1 6905 | . . 3 ⊢ (𝑥 = 𝑆 → (𝑥‘𝑁) = (𝑆‘𝑁)) | |
| 4 | strfvnd.c | . . . 4 ⊢ 𝐸 = Slot 𝑁 | |
| 5 | df-slot 17219 | . . . 4 ⊢ Slot 𝑁 = (𝑥 ∈ V ↦ (𝑥‘𝑁)) | |
| 6 | 4, 5 | eqtri 2765 | . . 3 ⊢ 𝐸 = (𝑥 ∈ V ↦ (𝑥‘𝑁)) |
| 7 | fvex 6919 | . . 3 ⊢ (𝑆‘𝑁) ∈ V | |
| 8 | 3, 6, 7 | fvmpt 7016 | . 2 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) = (𝑆‘𝑁)) |
| 9 | 1, 2, 8 | 3syl 18 | 1 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘𝑁)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ↦ cmpt 5225 ‘cfv 6561 Slot cslot 17218 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-slot 17219 |
| This theorem is referenced by: strfvn 17223 strfvss 17224 strndxid 17235 setsidvald 17236 strfvd 17237 strfv2d 17238 setsid 17244 setsnid 17245 setsnidOLD 17246 estrreslem1 18181 edgfndxid 29008 bj-endbase 37317 bj-endcomp 37318 |
| Copyright terms: Public domain | W3C validator |