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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 6733 | . . . . 5 ⊢ (𝑆 ∈ V → ( I ‘𝑆) = 𝑆) | |
3 | 2 | eqcomd 2824 | . . . 4 ⊢ (𝑆 ∈ V → 𝑆 = ( I ‘𝑆)) |
4 | 3 | fveq2d 6667 | . . 3 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) = (𝐸‘( I ‘𝑆))) |
5 | strfvi.e | . . . . 5 ⊢ 𝐸 = Slot 𝑁 | |
6 | 5 | str0 16523 | . . . 4 ⊢ ∅ = (𝐸‘∅) |
7 | fvprc 6656 | . . . 4 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = ∅) | |
8 | fvprc 6656 | . . . . 5 ⊢ (¬ 𝑆 ∈ V → ( I ‘𝑆) = ∅) | |
9 | 8 | fveq2d 6667 | . . . 4 ⊢ (¬ 𝑆 ∈ V → (𝐸‘( I ‘𝑆)) = (𝐸‘∅)) |
10 | 6, 7, 9 | 3eqtr4a 2879 | . . 3 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = (𝐸‘( I ‘𝑆))) |
11 | 4, 10 | pm2.61i 183 | . 2 ⊢ (𝐸‘𝑆) = (𝐸‘( I ‘𝑆)) |
12 | 1, 11 | eqtri 2841 | 1 ⊢ 𝑋 = (𝐸‘( I ‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∅c0 4288 I cid 5452 ‘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-pow 5257 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: rlmscaf 19909 islidl 19912 lidlrsppropd 19931 rspsn 19955 ply1tmcl 20368 ply1scltm 20377 ply1sclf 20381 ply1scl0 20386 ply1scl1 20388 nrgtrg 23226 |
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