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Mirrors > Home > MPE Home > Th. List > strfvi | Structured version Visualization version GIF version |
Description: Structure slot extractors cannot distinguish between proper classes and ∅, so they can be protected using the identity function. (Contributed by Stefan O'Rear, 21-Mar-2015.) |
Ref | Expression |
---|---|
strfvi.e | ⊢ 𝐸 = Slot 𝑁 |
strfvi.x | ⊢ 𝑋 = (𝐸‘𝑆) |
Ref | Expression |
---|---|
strfvi | ⊢ 𝑋 = (𝐸‘( I ‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strfvi.x | . 2 ⊢ 𝑋 = (𝐸‘𝑆) | |
2 | fvi 6165 | . . . . 5 ⊢ (𝑆 ∈ V → ( I ‘𝑆) = 𝑆) | |
3 | 2 | eqcomd 2616 | . . . 4 ⊢ (𝑆 ∈ V → 𝑆 = ( I ‘𝑆)) |
4 | 3 | fveq2d 6107 | . . 3 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) = (𝐸‘( I ‘𝑆))) |
5 | strfvi.e | . . . . 5 ⊢ 𝐸 = Slot 𝑁 | |
6 | 5 | str0 15739 | . . . 4 ⊢ ∅ = (𝐸‘∅) |
7 | fvprc 6097 | . . . 4 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = ∅) | |
8 | fvprc 6097 | . . . . 5 ⊢ (¬ 𝑆 ∈ V → ( I ‘𝑆) = ∅) | |
9 | 8 | fveq2d 6107 | . . . 4 ⊢ (¬ 𝑆 ∈ V → (𝐸‘( I ‘𝑆)) = (𝐸‘∅)) |
10 | 6, 7, 9 | 3eqtr4a 2670 | . . 3 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = (𝐸‘( I ‘𝑆))) |
11 | 4, 10 | pm2.61i 175 | . 2 ⊢ (𝐸‘𝑆) = (𝐸‘( I ‘𝑆)) |
12 | 1, 11 | eqtri 2632 | 1 ⊢ 𝑋 = (𝐸‘( I ‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 I cid 4948 ‘cfv 5804 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-8 1979 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-pow 4769 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: rlmscaf 19029 islidl 19032 lidlrsppropd 19051 rspsn 19075 ply1tmcl 19463 ply1scltm 19472 ply1sclf 19476 ply1scl0 19481 ply1scl1 19483 nrgtrg 22304 |
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