![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > op2ndd | 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 5534 | . 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 6171 | . 2 ⊢ (2nd ‘〈𝐴, 𝐵〉) = 𝐵 |
5 | 1, 4 | eqtrdi 2238 | 1 ⊢ (𝐶 = 〈𝐴, 𝐵〉 → (2nd ‘𝐶) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 Vcvv 2752 〈cop 3610 ‘cfv 5235 2nd c2nd 6163 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-iota 5196 df-fun 5237 df-fv 5243 df-2nd 6165 |
This theorem is referenced by: xp2nd 6190 sbcopeq1a 6211 csbopeq1a 6212 eloprabi 6220 mpomptsx 6221 dmmpossx 6223 fmpox 6224 fmpoco 6240 df2nd2 6244 xporderlem 6255 xpf1o 6871 frecuzrdgtcl 10442 frecuzrdgfunlem 10449 fisumcom2 11477 fprodcom2fi 11665 txbas 14210 cnmpt2nd 14241 txhmeo 14271 |
Copyright terms: Public domain | W3C validator |