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Mirrors > Home > MPE Home > Th. List > elbasov | Structured version Visualization version GIF version |
Description: Utility theorem: reverse closure for any structure defined as a two-argument function. (Contributed by Mario Carneiro, 3-Oct-2015.) |
Ref | Expression |
---|---|
elbasov.o | ⊢ Rel dom 𝑂 |
elbasov.s | ⊢ 𝑆 = (𝑋𝑂𝑌) |
elbasov.b | ⊢ 𝐵 = (Base‘𝑆) |
Ref | Expression |
---|---|
elbasov | ⊢ (𝐴 ∈ 𝐵 → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 4249 | . 2 ⊢ (𝐴 ∈ 𝐵 → ¬ 𝐵 = ∅) | |
2 | elbasov.s | . . . . 5 ⊢ 𝑆 = (𝑋𝑂𝑌) | |
3 | elbasov.o | . . . . . 6 ⊢ Rel dom 𝑂 | |
4 | 3 | ovprc 7173 | . . . . 5 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋𝑂𝑌) = ∅) |
5 | 2, 4 | syl5eq 2845 | . . . 4 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝑆 = ∅) |
6 | 5 | fveq2d 6649 | . . 3 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → (Base‘𝑆) = (Base‘∅)) |
7 | elbasov.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
8 | base0 16528 | . . 3 ⊢ ∅ = (Base‘∅) | |
9 | 6, 7, 8 | 3eqtr4g 2858 | . 2 ⊢ (¬ (𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝐵 = ∅) |
10 | 1, 9 | nsyl2 143 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝑋 ∈ V ∧ 𝑌 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 dom cdm 5519 Rel wrel 5524 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 |
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-ov 7138 df-slot 16479 df-base 16481 |
This theorem is referenced by: strov2rcl 16538 psrelbas 20617 psraddcl 20621 psrmulcllem 20625 psrvscafval 20628 psrvscacl 20631 resspsradd 20654 resspsrmul 20655 cphsubrglem 23782 mdegcl 24670 |
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