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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 473 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V) | |
2 | 1 | con3i 150 | . 2 ⊢ (¬ 𝐴 ∈ V → ¬ (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
3 | ovprc1.1 | . . 3 ⊢ Rel dom 𝐹 | |
4 | 3 | ovprc 6683 | . 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 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 dom cdm 5114 Rel wrel 5119 (class class class)co 6650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-dm 5124 df-iota 5851 df-fv 5896 df-ov 6653 |
This theorem is referenced by: mapdom2 8131 setsnid 15915 ressbas 15930 resslem 15933 ressinbas 15936 ressress 15938 oduval 17130 oduleval 17131 gsum0 17278 oppgval 17777 oppgplusfval 17778 mgpval 18492 opprval 18624 srasca 19181 rlmsca2 19201 resspsrbas 19415 mpfrcl 19518 psrbaspropd 19605 mplbaspropd 19607 evl1fval1 19695 dsmmval 20078 dsmmbas2 20081 dsmmfi 20082 qtopres 21501 fgabs 21683 tnglem 22444 tngds 22452 tchval 23017 resvsca 29830 resvlem 29831 mapco2g 37277 mzpmfp 37310 mendbas 37754 |
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