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Mirrors > Home > MPE Home > Th. List > strfvd | Structured version Visualization version GIF version |
Description: Deduction version of strfv 17176. (Contributed by Mario Carneiro, 15-Nov-2014.) |
Ref | Expression |
---|---|
strfvd.e | ⊢ 𝐸 = Slot (𝐸‘ndx) |
strfvd.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
strfvd.f | ⊢ (𝜑 → Fun 𝑆) |
strfvd.n | ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆) |
Ref | Expression |
---|---|
strfvd | ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strfvd.e | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) | |
2 | strfvd.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
3 | 1, 2 | strfvnd 17157 | . 2 ⊢ (𝜑 → (𝐸‘𝑆) = (𝑆‘(𝐸‘ndx))) |
4 | strfvd.f | . . 3 ⊢ (𝜑 → Fun 𝑆) | |
5 | strfvd.n | . . 3 ⊢ (𝜑 → 〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆) | |
6 | funopfv 6948 | . . 3 ⊢ (Fun 𝑆 → (〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆 → (𝑆‘(𝐸‘ndx)) = 𝐶)) | |
7 | 4, 5, 6 | sylc 65 | . 2 ⊢ (𝜑 → (𝑆‘(𝐸‘ndx)) = 𝐶) |
8 | 3, 7 | eqtr2d 2766 | 1 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 〈cop 4636 Fun wfun 6543 ‘cfv 6549 Slot cslot 17153 ndxcnx 17165 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 df-slot 17154 |
This theorem is referenced by: strssd 17178 |
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