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| Mirrors > Home > MPE Home > Th. List > op2ndd | Structured version Visualization version GIF version | ||
| Description: Extract the second member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.) |
| Ref | Expression |
|---|---|
| op1st.1 | ⊢ 𝐴 ∈ V |
| op1st.2 | ⊢ 𝐵 ∈ V |
| Ref | Expression |
|---|---|
| op2ndd | ⊢ (𝐶 = 〈𝐴, 𝐵〉 → (2nd ‘𝐶) = 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fveq2 6906 | . 2 ⊢ (𝐶 = 〈𝐴, 𝐵〉 → (2nd ‘𝐶) = (2nd ‘〈𝐴, 𝐵〉)) | |
| 2 | op1st.1 | . . 3 ⊢ 𝐴 ∈ V | |
| 3 | op1st.2 | . . 3 ⊢ 𝐵 ∈ V | |
| 4 | 2, 3 | op2nd 8023 | . 2 ⊢ (2nd ‘〈𝐴, 𝐵〉) = 𝐵 |
| 5 | 1, 4 | eqtrdi 2793 | 1 ⊢ (𝐶 = 〈𝐴, 𝐵〉 → (2nd ‘𝐶) = 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 ‘cfv 6561 2nd c2nd 8013 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 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-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-2nd 8015 |
| This theorem is referenced by: 2nd2val 8043 xp2nd 8047 sbcopeq1a 8074 csbopeq1a 8075 eloprabi 8088 mpomptsx 8089 dmmpossx 8091 fmpox 8092 ovmptss 8118 fmpoco 8120 df2nd2 8124 frxp 8151 xporderlem 8152 fnwelem 8156 fimaproj 8160 xpord2lem 8167 naddcllem 8714 xpf1o 9179 mapunen 9186 xpwdomg 9625 hsmexlem2 10467 nqereu 10969 uzrdgfni 13999 fsumcom2 15810 fprodcom2 16020 qredeu 16695 comfeq 17749 isfuncd 17910 cofucl 17933 funcres2b 17942 funcpropd 17947 xpcco2nd 18230 xpccatid 18233 1stf2 18238 2ndf2 18241 1stfcl 18242 2ndfcl 18243 prf2fval 18246 prfcl 18248 evlf2 18263 evlfcl 18267 curf12 18272 curf1cl 18273 curf2 18274 curfcl 18277 hof2fval 18300 hofcl 18304 txbas 23575 cnmpt2nd 23677 txhmeo 23811 ptuncnv 23815 ptunhmeo 23816 xpstopnlem1 23817 xkohmeo 23823 prdstmdd 24132 ucnimalem 24289 fmucndlem 24300 fsum2cn 24895 ovoliunlem1 25537 2sqreuop 27506 2sqreuopnn 27507 2sqreuoplt 27508 2sqreuopltb 27509 2sqreuopnnlt 27510 2sqreuopnnltb 27511 noseqrdgfn 28312 wlkl0 30386 fcnvgreu 32683 fsumiunle 32831 gsummpt2co 33051 gsumhashmul 33064 gsumwrd2dccatlem 33069 gsumwrd2dccat 33070 elrgspnlem2 33247 elrgspnsubrunlem2 33252 esumiun 34095 eulerpartlemgs2 34382 hgt750lemb 34671 satfv1 35368 satefvfmla0 35423 msubrsub 35531 msubco 35536 msubvrs 35565 filnetlem4 36382 finixpnum 37612 poimirlem4 37631 poimirlem15 37642 poimirlem20 37647 poimirlem26 37653 heicant 37662 heiborlem4 37821 heiborlem6 37823 dicelvalN 41180 aks6d1c2p1 42119 aks6d1c3 42124 aks6d1c4 42125 aks6d1c6lem2 42172 aks6d1c6lem4 42174 aks6d1c7lem1 42181 fmpocos 42275 rmxypairf1o 42923 unxpwdom3 43107 fgraphxp 43216 elcnvlem 43614 dvnprodlem2 45962 etransclem46 46295 ovnsubaddlem1 46585 gpgvtxel2 48006 gpgvtx0 48008 gpgvtx1 48009 gpgedgvtx0 48019 gpgedgvtx1 48020 gpgvtxedg0 48021 gpgvtxedg1 48022 uspgrsprf 48062 uspgrsprf1 48063 dmmpossx2 48253 lmod1zr 48410 2arymaptf 48573 rrx2plordisom 48644 funcf2lem 48914 tposcurf1 48999 |
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