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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 4068 | . 2 ⊢ (𝑋 ∈ 𝐵 → ¬ 𝐵 = ∅) | |
2 | elbasfv.s | . . . . 5 ⊢ 𝑆 = (𝐹‘𝑍) | |
3 | fvprc 6326 | . . . . 5 ⊢ (¬ 𝑍 ∈ V → (𝐹‘𝑍) = ∅) | |
4 | 2, 3 | syl5eq 2817 | . . . 4 ⊢ (¬ 𝑍 ∈ V → 𝑆 = ∅) |
5 | 4 | fveq2d 6336 | . . 3 ⊢ (¬ 𝑍 ∈ V → (Base‘𝑆) = (Base‘∅)) |
6 | elbasfv.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
7 | base0 16119 | . . 3 ⊢ ∅ = (Base‘∅) | |
8 | 5, 6, 7 | 3eqtr4g 2830 | . 2 ⊢ (¬ 𝑍 ∈ V → 𝐵 = ∅) |
9 | 1, 8 | nsyl2 144 | 1 ⊢ (𝑋 ∈ 𝐵 → 𝑍 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1631 ∈ wcel 2145 Vcvv 3351 ∅c0 4063 ‘cfv 6031 Basecbs 16064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-iota 5994 df-fun 6033 df-fv 6039 df-slot 16068 df-base 16070 |
This theorem is referenced by: frmdelbas 17598 symginv 18029 symggen 18097 psgneu 18133 psgnpmtr 18137 coe1sfi 19798 frgpcyg 20137 lindfind 20372 q1pval 24133 r1pval 24136 |
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