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Mirrors > Home > MPE Home > Th. List > strov2rcl | Structured version Visualization version GIF version |
Description: Partial reverse closure for any structure defined as a two-argument function. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Proof shortened by AV, 2-Dec-2019.) |
Ref | Expression |
---|---|
strov2rcl.s | ⊢ 𝑆 = (𝐼𝐹𝑅) |
strov2rcl.b | ⊢ 𝐵 = (Base‘𝑆) |
strov2rcl.f | ⊢ Rel dom 𝐹 |
Ref | Expression |
---|---|
strov2rcl | ⊢ (𝑋 ∈ 𝐵 → 𝐼 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strov2rcl.f | . . 3 ⊢ Rel dom 𝐹 | |
2 | strov2rcl.s | . . 3 ⊢ 𝑆 = (𝐼𝐹𝑅) | |
3 | strov2rcl.b | . . 3 ⊢ 𝐵 = (Base‘𝑆) | |
4 | 1, 2, 3 | elbasov 16533 | . 2 ⊢ (𝑋 ∈ 𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V)) |
5 | 4 | simpld 495 | 1 ⊢ (𝑋 ∈ 𝐵 → 𝐼 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 dom cdm 5548 Rel wrel 5553 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-slot 16475 df-base 16477 |
This theorem is referenced by: mplrcl 20198 psropprmul 20334 dsmmbas2 20809 frlmrcl 20829 |
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