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Mirrors > Home > MPE Home > Th. List > strfvnd | Structured version Visualization version GIF version |
Description: Deduction version of strfvn 17220. (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 3499 | . 2 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) | |
3 | fveq1 6906 | . . 3 ⊢ (𝑥 = 𝑆 → (𝑥‘𝑁) = (𝑆‘𝑁)) | |
4 | strfvnd.c | . . . 4 ⊢ 𝐸 = Slot 𝑁 | |
5 | df-slot 17216 | . . . 4 ⊢ Slot 𝑁 = (𝑥 ∈ V ↦ (𝑥‘𝑁)) | |
6 | 4, 5 | eqtri 2763 | . . 3 ⊢ 𝐸 = (𝑥 ∈ V ↦ (𝑥‘𝑁)) |
7 | fvex 6920 | . . 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 1537 ∈ wcel 2106 Vcvv 3478 ↦ cmpt 5231 ‘cfv 6563 Slot cslot 17215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-slot 17216 |
This theorem is referenced by: strfvn 17220 strfvss 17221 strndxid 17232 setsidvald 17233 setsidvaldOLD 17234 strfvd 17235 strfv2d 17236 setsid 17242 setsnid 17243 setsnidOLD 17244 estrreslem1 18192 edgfndxid 29023 bj-endbase 37299 bj-endcomp 37300 |
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