Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ovmpoa | Structured version Visualization version GIF version |
Description: Value of an operation given by a maps-to rule. (Contributed by NM, 19-Dec-2013.) |
Ref | Expression |
---|---|
ovmpoga.1 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆) |
ovmpoga.2 | ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) |
ovmpoa.4 | ⊢ 𝑆 ∈ V |
Ref | Expression |
---|---|
ovmpoa | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝐹𝐵) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovmpoa.4 | . 2 ⊢ 𝑆 ∈ V | |
2 | ovmpoga.1 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆) | |
3 | ovmpoga.2 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) | |
4 | 2, 3 | ovmpoga 7304 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ V) → (𝐴𝐹𝐵) = 𝑆) |
5 | 1, 4 | mp3an3 1446 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝐹𝐵) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 (class class class)co 7156 ∈ cmpo 7158 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 |
This theorem is referenced by: 1st2val 7717 2nd2val 7718 cantnffval 9126 cantnfsuc 9133 fseqenlem1 9450 xaddval 12617 xmulval 12619 fzoval 13040 expval 13432 ccatfval 13925 splcl 14114 cshfn 14152 bpolylem 15402 ruclem1 15584 sadfval 15801 sadcp1 15804 smufval 15826 smupp1 15829 eucalgval2 15925 pcval 16181 pc0 16191 vdwapval 16309 pwsval 16759 xpsfval 16839 xpsval 16843 rescval 17097 isfunc 17134 isfull 17180 isfth 17184 natfval 17216 catcisolem 17366 xpchom 17430 1stfval 17441 2ndfval 17444 yonedalem3a 17524 yonedainv 17531 plusfval 17859 ismhm 17958 mulgval 18228 eqgfval 18328 isga 18421 subgga 18430 cayleylem1 18540 sylow1lem2 18724 isslw 18733 sylow2blem1 18745 sylow3lem1 18752 sylow3lem6 18757 frgpuptinv 18897 frgpup2 18902 isrhm 19473 scafval 19653 islmhm 19799 psrmulfval 20165 mplval 20208 ltbval 20252 mpfrcl 20298 evlsval 20299 evlval 20308 xrsdsval 20589 ipfval 20793 dsmmval 20878 matval 21020 submafval 21188 mdetfval 21195 minmar1fval 21255 txval 22172 xkoval 22195 hmeofval 22366 flffval 22597 qustgplem 22729 dscmet 23182 dscopn 23183 tngval 23248 nmofval 23323 nghmfval 23331 isnmhm 23355 htpyco1 23582 htpycc 23584 phtpycc 23595 reparphti 23601 pcoval 23615 pcohtpylem 23623 pcorevlem 23630 dyadval 24193 itg1addlem3 24299 itg1addlem4 24300 mbfi1fseqlem3 24318 mbfi1fseqlem4 24319 mbfi1fseqlem5 24320 mbfi1fseqlem6 24321 mdegfval 24656 quotval 24881 elqaalem2 24909 cxpval 25247 cxpcn3 25329 angval 25379 sgmval 25719 lgsval 25877 wwlksn 27615 wspthsn 27626 rusgrnumwwlklem 27749 clwwlkn 27804 2clwwlk 28126 numclwwlkovh0 28151 numclwwlkovq 28153 shsval 29089 sshjval 29127 faeval 31505 txsconnlem 32487 cvxsconn 32490 iscvm 32506 cvmliftlem5 32536 rngohomval 35257 rngoisoval 35270 rmxfval 39521 rmyfval 39522 mendplusg 39806 mendvsca 39811 addrval 40818 subrval 40819 mulvval 40820 sigarval 43127 ismgmhm 44070 dmatALTval 44475 |
Copyright terms: Public domain | W3C validator |