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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 6103 | . 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 7068 | . 2 ⊢ (2nd ‘〈𝐴, 𝐵〉) = 𝐵 |
5 | 1, 4 | syl6eq 2660 | 1 ⊢ (𝐶 = 〈𝐴, 𝐵〉 → (2nd ‘𝐶) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 ‘cfv 5804 2nd c2nd 7058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-2nd 7060 |
This theorem is referenced by: 2nd2val 7086 xp2nd 7090 sbcopeq1a 7111 csbopeq1a 7112 eloprabi 7121 mpt2mptsx 7122 dmmpt2ssx 7124 fmpt2x 7125 ovmptss 7145 fmpt2co 7147 df2nd2 7151 frxp 7174 xporderlem 7175 fnwelem 7179 xpf1o 8007 mapunen 8014 xpwdomg 8373 hsmexlem2 9132 nqereu 9630 uzrdgfni 12619 fsumcom2 14347 fsumcom2OLD 14348 fprodcom2 14553 fprodcom2OLD 14554 qredeu 15210 comfeq 16189 isfuncd 16348 cofucl 16371 funcres2b 16380 funcpropd 16383 xpcco2nd 16648 xpccatid 16651 1stf2 16656 2ndf2 16659 1stfcl 16660 2ndfcl 16661 prf2fval 16664 prfcl 16666 evlf2 16681 evlfcl 16685 curf12 16690 curf1cl 16691 curf2 16692 curfcl 16695 hof2fval 16718 hofcl 16722 txbas 21180 cnmpt2nd 21282 txhmeo 21416 ptuncnv 21420 ptunhmeo 21421 xpstopnlem1 21422 xkohmeo 21428 prdstmdd 21737 ucnimalem 21894 fmucndlem 21905 fsum2cn 22482 ovoliunlem1 23077 usgrac 25880 edgss 25881 fcnvgreu 28855 gsummpt2co 29111 fimaproj 29228 esumiun 29483 eulerpartlemgs2 29769 msubrsub 30677 msubco 30682 msubvrs 30711 filnetlem4 31546 finixpnum 32564 poimirlem4 32583 poimirlem15 32594 poimirlem20 32599 poimirlem26 32605 heicant 32614 heiborlem4 32783 heiborlem6 32785 dicelvalN 35485 rmxypairf1o 36494 unxpwdom3 36683 fgraphxp 36808 elcnvlem 36926 dvnprodlem2 38837 etransclem46 39173 ovnsubaddlem1 39460 dmmpt2ssx2 41908 lmod1zr 42076 |
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