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Mirrors > Home > MPE Home > Th. List > strfvnd | Structured version Visualization version GIF version |
Description: Deduction version of strfvn 16493. (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 3510 | . 2 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) | |
3 | fveq1 6662 | . . 3 ⊢ (𝑥 = 𝑆 → (𝑥‘𝑁) = (𝑆‘𝑁)) | |
4 | strfvnd.c | . . . 4 ⊢ 𝐸 = Slot 𝑁 | |
5 | df-slot 16475 | . . . 4 ⊢ Slot 𝑁 = (𝑥 ∈ V ↦ (𝑥‘𝑁)) | |
6 | 4, 5 | eqtri 2841 | . . 3 ⊢ 𝐸 = (𝑥 ∈ V ↦ (𝑥‘𝑁)) |
7 | fvex 6676 | . . 3 ⊢ (𝑆‘𝑁) ∈ V | |
8 | 3, 6, 7 | fvmpt 6761 | . 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 1528 ∈ wcel 2105 Vcvv 3492 ↦ cmpt 5137 ‘cfv 6348 Slot cslot 16470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-slot 16475 |
This theorem is referenced by: strfvn 16493 strfvss 16494 strndxid 16498 setsidvald 16502 strfvd 16516 strfv2d 16517 setsid 16526 setsnid 16527 |
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