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Mirrors > Home > MPE Home > Th. List > elbasfv | Structured version Visualization version GIF version |
Description: Utility theorem: reverse closure for any structure defined as a function. (Contributed by Stefan O'Rear, 24-Aug-2015.) |
Ref | Expression |
---|---|
elbasfv.s | ⊢ 𝑆 = (𝐹‘𝑍) |
elbasfv.b | ⊢ 𝐵 = (Base‘𝑆) |
Ref | Expression |
---|---|
elbasfv | ⊢ (𝑋 ∈ 𝐵 → 𝑍 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 4262 | . 2 ⊢ (𝑋 ∈ 𝐵 → ¬ 𝐵 = ∅) | |
2 | elbasfv.s | . . . . 5 ⊢ 𝑆 = (𝐹‘𝑍) | |
3 | fvprc 6727 | . . . . 5 ⊢ (¬ 𝑍 ∈ V → (𝐹‘𝑍) = ∅) | |
4 | 2, 3 | eqtrid 2790 | . . . 4 ⊢ (¬ 𝑍 ∈ V → 𝑆 = ∅) |
5 | 4 | fveq2d 6739 | . . 3 ⊢ (¬ 𝑍 ∈ V → (Base‘𝑆) = (Base‘∅)) |
6 | elbasfv.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
7 | base0 16789 | . . 3 ⊢ ∅ = (Base‘∅) | |
8 | 5, 6, 7 | 3eqtr4g 2804 | . 2 ⊢ (¬ 𝑍 ∈ V → 𝐵 = ∅) |
9 | 1, 8 | nsyl2 143 | 1 ⊢ (𝑋 ∈ 𝐵 → 𝑍 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1543 ∈ wcel 2111 Vcvv 3420 ∅c0 4251 ‘cfv 6397 Basecbs 16784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-cnex 10809 ax-1cn 10811 ax-addcl 10813 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-iun 4920 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-pred 6175 df-ord 6233 df-on 6234 df-lim 6235 df-suc 6236 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-ov 7234 df-om 7663 df-wrecs 8067 df-recs 8128 df-rdg 8166 df-nn 11855 df-slot 16759 df-ndx 16769 df-base 16785 |
This theorem is referenced by: frmdelbas 18304 symginv 18818 symggen 18886 psgneu 18922 psgnpmtr 18926 frgpcyg 20562 lindfind 20802 coe1sfi 21158 q1pval 25075 r1pval 25078 symgsubg 31099 |
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