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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 4301 | . 2 ⊢ (𝑋 ∈ 𝐵 → ¬ 𝐵 = ∅) | |
2 | elbasfv.s | . . . . 5 ⊢ 𝑆 = (𝐹‘𝑍) | |
3 | fvprc 6665 | . . . . 5 ⊢ (¬ 𝑍 ∈ V → (𝐹‘𝑍) = ∅) | |
4 | 2, 3 | syl5eq 2870 | . . . 4 ⊢ (¬ 𝑍 ∈ V → 𝑆 = ∅) |
5 | 4 | fveq2d 6676 | . . 3 ⊢ (¬ 𝑍 ∈ V → (Base‘𝑆) = (Base‘∅)) |
6 | elbasfv.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
7 | base0 16538 | . . 3 ⊢ ∅ = (Base‘∅) | |
8 | 5, 6, 7 | 3eqtr4g 2883 | . 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 1537 ∈ wcel 2114 Vcvv 3496 ∅c0 4293 ‘cfv 6357 Basecbs 16485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-slot 16489 df-base 16491 |
This theorem is referenced by: frmdelbas 18020 symginv 18532 symggen 18600 psgneu 18636 psgnpmtr 18640 coe1sfi 20383 frgpcyg 20722 lindfind 20962 q1pval 24749 r1pval 24752 symgsubg 30733 |
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