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Mirrors > Home > MPE Home > Th. List > ovmpt2 | Structured version Visualization version GIF version |
Description: Value of an operation given by a maps-to rule. Special case. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.) |
Ref | Expression |
---|---|
ovmpt2g.1 | ⊢ (𝑥 = 𝐴 → 𝑅 = 𝐺) |
ovmpt2g.2 | ⊢ (𝑦 = 𝐵 → 𝐺 = 𝑆) |
ovmpt2g.3 | ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) |
ovmpt2.4 | ⊢ 𝑆 ∈ V |
Ref | Expression |
---|---|
ovmpt2 | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝐹𝐵) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovmpt2.4 | . 2 ⊢ 𝑆 ∈ V | |
2 | ovmpt2g.1 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑅 = 𝐺) | |
3 | ovmpt2g.2 | . . 3 ⊢ (𝑦 = 𝐵 → 𝐺 = 𝑆) | |
4 | ovmpt2g.3 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) | |
5 | 2, 3, 4 | ovmpt2g 6795 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ V) → (𝐴𝐹𝐵) = 𝑆) |
6 | 1, 5 | mp3an3 1413 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝐹𝐵) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 (class class class)co 6650 ↦ cmpt2 6652 |
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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 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-sbc 3436 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-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 |
This theorem is referenced by: seqomlem1 7545 seqomlem4 7548 oav 7591 omv 7592 oev 7594 iunfictbso 8937 fin23lem12 9153 axdc4lem 9277 axcclem 9279 addpipq2 9758 mulpipq2 9761 subval 10272 divval 10687 cnref1o 11827 ixxval 12183 fzval 12328 modval 12670 om2uzrdg 12755 uzrdgsuci 12759 axdc4uzlem 12782 seqval 12812 seqp1 12816 bcval 13091 cnrecnv 13905 risefacval 14739 fallfacval 14740 gcdval 15218 lcmval 15305 imasvscafn 16197 imasvscaval 16198 grpsubval 17465 isghm 17660 lactghmga 17824 efgmval 18125 efgtval 18136 frgpup3lem 18190 dvrval 18685 psrvsca 19391 frlmval 20092 mat1comp 20246 mamulid 20247 mamurid 20248 madufval 20443 xkococnlem 21462 xkococn 21463 cnextval 21865 dscmet 22377 cncfval 22691 htpycom 22775 htpyid 22776 phtpycom 22787 phtpyid 22788 logbval 24504 isismt 25429 wwlks2onv 26847 grpodivval 27389 ipval 27558 lnoval 27607 nmoofval 27617 bloval 27636 0ofval 27642 ajfval 27664 hvsubval 27873 hosmval 28594 hommval 28595 hodmval 28596 hfsmval 28597 hfmmval 28598 kbfval 28811 opsqrlem3 29001 dpval 29597 xdivval 29627 smatrcl 29862 smatlem 29863 mdetpmtr12 29891 fvproj 29899 pstmfval 29939 sxval 30253 ismbfm 30314 dya2iocival 30335 sitgval 30394 sitmval 30411 oddpwdcv 30417 ballotlemgval 30585 vtsval 30715 cvmlift2lem4 31288 icoreval 33201 metf1o 33551 heiborlem3 33612 heiborlem6 33615 heiborlem8 33617 heibor 33620 ldualvs 34424 tendopl 36064 cdlemkuu 36183 dvavsca 36305 dvhvaddval 36379 dvhvscaval 36388 hlhilipval 37241 |
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