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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 6715 | . . . . 5 ⊢ (𝑆 ∈ V → ( I ‘𝑆) = 𝑆) | |
3 | 2 | eqcomd 2804 | . . . 4 ⊢ (𝑆 ∈ V → 𝑆 = ( I ‘𝑆)) |
4 | 3 | fveq2d 6649 | . . 3 ⊢ (𝑆 ∈ V → (𝐸‘𝑆) = (𝐸‘( I ‘𝑆))) |
5 | strfvi.e | . . . . 5 ⊢ 𝐸 = Slot 𝑁 | |
6 | 5 | str0 16527 | . . . 4 ⊢ ∅ = (𝐸‘∅) |
7 | fvprc 6638 | . . . 4 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = ∅) | |
8 | fvprc 6638 | . . . . 5 ⊢ (¬ 𝑆 ∈ V → ( I ‘𝑆) = ∅) | |
9 | 8 | fveq2d 6649 | . . . 4 ⊢ (¬ 𝑆 ∈ V → (𝐸‘( I ‘𝑆)) = (𝐸‘∅)) |
10 | 6, 7, 9 | 3eqtr4a 2859 | . . 3 ⊢ (¬ 𝑆 ∈ V → (𝐸‘𝑆) = (𝐸‘( I ‘𝑆))) |
11 | 4, 10 | pm2.61i 185 | . 2 ⊢ (𝐸‘𝑆) = (𝐸‘( I ‘𝑆)) |
12 | 1, 11 | eqtri 2821 | 1 ⊢ 𝑋 = (𝐸‘( I ‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 I cid 5424 ‘cfv 6324 Slot cslot 16474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-slot 16479 |
This theorem is referenced by: rlmscaf 19974 islidl 19977 lidlrsppropd 19996 rspsn 20020 ply1tmcl 20901 ply1scltm 20910 ply1sclf 20914 ply1scl0 20919 ply1scl1 20921 nrgtrg 23296 |
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