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Mirrors > Home > MPE Home > Th. List > strfvd | Structured version Visualization version GIF version |
Description: Deduction version of strfv 15735. (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 15710 | . 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 6145 | . . 3 ⊢ (Fun 𝑆 → (〈(𝐸‘ndx), 𝐶〉 ∈ 𝑆 → (𝑆‘(𝐸‘ndx)) = 𝐶)) | |
7 | 4, 5, 6 | sylc 63 | . 2 ⊢ (𝜑 → (𝑆‘(𝐸‘ndx)) = 𝐶) |
8 | 3, 7 | eqtr2d 2645 | 1 ⊢ (𝜑 → 𝐶 = (𝐸‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 〈cop 4131 Fun wfun 5798 ‘cfv 5804 ndxcnx 15692 Slot cslot 15694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-slot 15699 |
This theorem is referenced by: strssd 15737 |
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