| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > ovres | Structured version Visualization version GIF version | ||
| Description: The value of a restricted operation. (Contributed by FL, 10-Nov-2006.) |
| Ref | Expression |
|---|---|
| ovres | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐹𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | opelxpi 5722 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷)) | |
| 2 | 1 | fvresd 6926 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((𝐹 ↾ (𝐶 × 𝐷))‘〈𝐴, 𝐵〉) = (𝐹‘〈𝐴, 𝐵〉)) |
| 3 | df-ov 7434 | . 2 ⊢ (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = ((𝐹 ↾ (𝐶 × 𝐷))‘〈𝐴, 𝐵〉) | |
| 4 | df-ov 7434 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
| 5 | 2, 3, 4 | 3eqtr4g 2802 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐹𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 〈cop 4632 × cxp 5683 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-res 5697 df-iota 6514 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: ovresd 7600 oprres 7601 oprssov 7602 ofmresval 7713 cantnfval2 9709 mulnzcnf 11909 prdsdsval3 17530 mgmsscl 18658 frmdplusg 18867 frmdadd 18868 grpissubg 19164 gaid 19317 gass 19319 gasubg 19320 rnghmresel 20620 rnghmsscmap2 20629 rnghmsscmap 20630 rnghmsubcsetclem2 20632 rngcifuestrc 20639 rhmresel 20649 rhmsscmap2 20658 rhmsscmap 20659 rhmsubcsetclem2 20661 rhmsscrnghm 20665 rhmsubcrngclem2 20667 rhmsubclem4 20688 mplsubrglem 22024 mamures 22401 mdetrlin 22608 mdetrsca 22609 pmatcollpw3lem 22789 tsmsxplem1 24161 tsmsxplem2 24162 xmetres2 24371 ressprdsds 24381 blres 24441 xmetresbl 24447 mscl 24471 xmscl 24472 xmsge0 24473 xmseq0 24474 nmfval0 24603 nmval2 24605 isngp3 24611 ngpds 24617 ngpocelbl 24725 xrsdsre 24832 divcnOLD 24890 divcn 24892 cncfmet 24935 cfilresi 25329 cfilres 25330 mpodvdsmulf1o 27237 dvdsmulf1o 27239 sspgval 30748 sspsval 30750 sspmlem 30751 hhssabloilem 31280 hhssabloi 31281 hhssnv 31283 hhssmetdval 31296 raddcn 33928 xrge0pluscn 33939 cvmlift2lem9 35316 icoreval 37354 icoreelrnab 37355 equivbnd2 37799 ismtyres 37815 iccbnd 37847 exidreslem 37884 divrngcl 37964 isdrngo2 37965 ofoafo 43369 ofoacl 43370 naddcnfcl 43378 fuco11b 49032 |
| Copyright terms: Public domain | W3C validator |