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Mirrors > Home > MPE Home > Th. List > strfvd | Structured version Visualization version GIF version |
Description: Deduction version of strfv 16314. (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 16285 | . 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 6496 | . . 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 2815 | 1 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 〈cop 4404 Fun wfun 6131 ‘cfv 6137 ndxcnx 16263 Slot cslot 16265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-iota 6101 df-fun 6139 df-fv 6145 df-slot 16270 |
This theorem is referenced by: strssd 16316 |
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