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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 5286 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷)) | |
2 | fvres 6346 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ (𝐶 × 𝐷) → ((𝐹 ↾ (𝐶 × 𝐷))‘〈𝐴, 𝐵〉) = (𝐹‘〈𝐴, 𝐵〉)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((𝐹 ↾ (𝐶 × 𝐷))‘〈𝐴, 𝐵〉) = (𝐹‘〈𝐴, 𝐵〉)) |
4 | df-ov 6794 | . 2 ⊢ (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = ((𝐹 ↾ (𝐶 × 𝐷))‘〈𝐴, 𝐵〉) | |
5 | df-ov 6794 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
6 | 3, 4, 5 | 3eqtr4g 2830 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐹𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 〈cop 4322 × cxp 5247 ↾ cres 5251 ‘cfv 6029 (class class class)co 6791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-xp 5255 df-res 5261 df-iota 5992 df-fv 6037 df-ov 6794 |
This theorem is referenced by: ovresd 6946 oprres 6947 oprssov 6948 ofmresval 7055 cantnfval2 8728 mulnzcnopr 10873 prdsdsval3 16346 frmdplusg 17592 frmdadd 17593 grpissubg 17815 gaid 17932 gass 17934 gasubg 17935 mplsubrglem 19647 mamures 20406 mdetrlin 20619 mdetrsca 20620 pmatcollpw3lem 20801 tsmsxplem1 22169 tsmsxplem2 22170 xmetres2 22379 ressprdsds 22389 blres 22449 xmetresbl 22455 mscl 22479 xmscl 22480 xmsge0 22481 xmseq0 22482 nmfval2 22608 nmval2 22609 isngp3 22615 ngpds 22621 ngpocelbl 22721 xrsdsre 22826 divcn 22884 cncfmet 22924 cfilresi 23305 cfilres 23306 dvdsmulf1o 25134 sspgval 27917 sspsval 27919 sspmlem 27920 hhssabloilem 28451 hhssabloi 28452 hhssnv 28454 hhssmetdval 28468 raddcn 30308 xrge0pluscn 30319 cvmlift2lem9 31624 icoreval 33531 icoreelrnab 33532 equivbnd2 33916 ismtyres 33932 iccbnd 33964 exidreslem 34001 divrngcl 34081 isdrngo2 34082 rnghmresel 42485 rnghmsscmap2 42494 rnghmsscmap 42495 rnghmsubcsetclem2 42497 rngcifuestrc 42518 rhmresel 42531 rhmsscmap2 42540 rhmsscmap 42541 rhmsubcsetclem2 42543 rhmsscrnghm 42547 rhmsubcrngclem2 42549 rhmsubclem4 42610 |
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