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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 5421 | . 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 6045 | . 2 ⊢ (2nd ‘〈𝐴, 𝐵〉) = 𝐵 |
5 | 1, 4 | syl6eq 2188 | 1 ⊢ (𝐶 = 〈𝐴, 𝐵〉 → (2nd ‘𝐶) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 Vcvv 2686 〈cop 3530 ‘cfv 5123 2nd c2nd 6037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fv 5131 df-2nd 6039 |
This theorem is referenced by: xp2nd 6064 sbcopeq1a 6085 csbopeq1a 6086 eloprabi 6094 mpomptsx 6095 dmmpossx 6097 fmpox 6098 fmpoco 6113 df2nd2 6117 xporderlem 6128 xpf1o 6738 frecuzrdgtcl 10185 frecuzrdgfunlem 10192 fisumcom2 11207 txbas 12427 cnmpt2nd 12458 txhmeo 12488 |
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