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Mirrors > Home > MPE Home > Th. List > ovprc1 | Structured version Visualization version GIF version |
Description: The value of an operation when the first argument is a proper class. (Contributed by NM, 16-Jun-2004.) |
Ref | Expression |
---|---|
ovprc1.1 | ⊢ Rel dom 𝐹 |
Ref | Expression |
---|---|
ovprc1 | ⊢ (¬ 𝐴 ∈ V → (𝐴𝐹𝐵) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 483 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V) | |
2 | 1 | con3i 157 | . 2 ⊢ (¬ 𝐴 ∈ V → ¬ (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
3 | ovprc1.1 | . . 3 ⊢ Rel dom 𝐹 | |
4 | 3 | ovprc 7183 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐹𝐵) = ∅) |
5 | 2, 4 | syl 17 | 1 ⊢ (¬ 𝐴 ∈ V → (𝐴𝐹𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∅c0 4288 dom cdm 5548 Rel wrel 5553 (class class class)co 7145 |
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-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-xp 5554 df-rel 5555 df-dm 5558 df-iota 6307 df-fv 6356 df-ov 7148 |
This theorem is referenced by: mapdom2 8676 setsnid 16527 ressbas 16542 resslem 16545 ressinbas 16548 ressress 16550 oduval 17728 oduleval 17729 gsum0 17882 oppgval 18413 oppgplusfval 18414 mgpval 19171 opprval 19303 srasca 19882 rlmsca2 19902 resspsrbas 20123 mpfrcl 20226 psrbaspropd 20331 mplbaspropd 20333 evl1fval1 20422 dsmmval 20806 dsmmfi 20810 qtopres 22234 fgabs 22415 tnglem 23176 tngds 23184 tcphval 23748 resvsca 30830 resvlem 30831 mapco2g 39189 mzpmfp 39222 mendbas 39662 efmndbas 43970 |
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