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Mirrors > Home > MPE Home > Th. List > op1std | Structured version Visualization version GIF version |
Description: Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
op1st.1 | ⊢ 𝐴 ∈ V |
op1st.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
op1std | ⊢ (𝐶 = 〈𝐴, 𝐵〉 → (1st ‘𝐶) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6906 | . 2 ⊢ (𝐶 = 〈𝐴, 𝐵〉 → (1st ‘𝐶) = (1st ‘〈𝐴, 𝐵〉)) | |
2 | op1st.1 | . . 3 ⊢ 𝐴 ∈ V | |
3 | op1st.2 | . . 3 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | op1st 8020 | . 2 ⊢ (1st ‘〈𝐴, 𝐵〉) = 𝐴 |
5 | 1, 4 | eqtrdi 2790 | 1 ⊢ (𝐶 = 〈𝐴, 𝐵〉 → (1st ‘𝐶) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 Vcvv 3477 〈cop 4636 ‘cfv 6562 1st c1st 8010 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-iota 6515 df-fun 6564 df-fv 6570 df-1st 8012 |
This theorem is referenced by: 1st2val 8040 xp1st 8044 sbcopeq1a 8072 csbopeq1a 8073 eloprabi 8086 mpomptsx 8087 dmmpossx 8089 fmpox 8090 ovmptss 8116 fmpoco 8118 df1st2 8121 fsplit 8140 frxp 8149 xporderlem 8150 fnwelem 8154 fimaproj 8158 xpord2lem 8165 naddcllem 8712 xpf1o 9177 mapunen 9184 xpwdomg 9622 hsmexlem2 10464 fsumcom2 15806 fprodcom2 16016 qredeu 16691 isfuncd 17915 cofucl 17938 funcres2b 17947 funcpropd 17953 xpcco1st 18239 xpccatid 18243 1stf1 18247 2ndf1 18250 1stfcl 18252 prf1 18255 prfcl 18258 prf1st 18259 prf2nd 18260 evlf1 18276 evlfcl 18278 curf1fval 18280 curf11 18282 curf1cl 18284 curfcl 18288 hof1fval 18309 txbas 23590 cnmpt1st 23691 txhmeo 23826 ptuncnv 23830 ptunhmeo 23831 xpstopnlem1 23832 xkohmeo 23838 prdstmdd 24147 ucnimalem 24304 fmucndlem 24315 fsum2cn 24908 ovoliunlem1 25550 lgsquadlem1 27438 lgsquadlem2 27439 2sqreuop 27520 2sqreuopnn 27521 2sqreuoplt 27522 2sqreuopltb 27523 2sqreuopnnlt 27524 2sqreuopnnltb 27525 clwlkclwwlkfolem 30035 wlkl0 30395 gsumhashmul 33046 gsumwrd2dccatlem 33051 gsumwrd2dccat 33052 elrgspnlem2 33232 eulerpartlemgs2 34361 hgt750lemb 34649 cvmliftlem15 35282 satfv1 35347 satfdmlem 35352 fmlasuc 35370 msubty 35511 msubco 35515 msubvrs 35544 filnetlem4 36363 finixpnum 37591 poimirlem4 37610 poimirlem15 37621 poimirlem20 37626 poimirlem26 37632 poimirlem28 37634 heicant 37641 dicelvalN 41160 aks6d1c2p1 42099 aks6d1c3 42104 aks6d1c4 42105 aks6d1c6lem2 42152 aks6d1c6lem4 42154 aks6d1c7lem1 42161 fmpocos 42253 rmxypairf1o 42899 unxpwdom3 43083 fgraphxp 43192 elcnvlem 43590 dvnprodlem2 45902 etransclem46 46235 ovnsubaddlem1 46525 gpgvtxedg0 47955 gpgvtxedg1 47956 gpgcubic 47969 gpg5nbgr3star 47971 dmmpossx2 48181 2arymaptf 48501 rrx2plordisom 48572 funcf2lem 48810 |
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