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| Mirrors > Home > MPE Home > Th. List > op2nda | Structured version Visualization version GIF version | ||
| Description: Extract the second member of an ordered pair. (See op1sta 6245 to extract the first member, op2ndb 6247 for an alternate version, and op2nd 8023 for the preferred version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
| Ref | Expression |
|---|---|
| cnvsn.1 | ⊢ 𝐴 ∈ V |
| cnvsn.2 | ⊢ 𝐵 ∈ V |
| Ref | Expression |
|---|---|
| op2nda | ⊢ ∪ ran {〈𝐴, 𝐵〉} = 𝐵 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cnvsn.1 | . . . 4 ⊢ 𝐴 ∈ V | |
| 2 | 1 | rnsnop 6244 | . . 3 ⊢ ran {〈𝐴, 𝐵〉} = {𝐵} |
| 3 | 2 | unieqi 4919 | . 2 ⊢ ∪ ran {〈𝐴, 𝐵〉} = ∪ {𝐵} |
| 4 | cnvsn.2 | . . 3 ⊢ 𝐵 ∈ V | |
| 5 | 4 | unisn 4926 | . 2 ⊢ ∪ {𝐵} = 𝐵 |
| 6 | 3, 5 | eqtri 2765 | 1 ⊢ ∪ ran {〈𝐴, 𝐵〉} = 𝐵 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 〈cop 4632 ∪ cuni 4907 ran crn 5686 |
| 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-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 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-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 |
| This theorem is referenced by: elxp4 7944 elxp5 7945 op2nd 8023 fo2nd 8035 f2ndres 8039 ixpsnf1o 8978 xpassen 9106 xpdom2 9107 |
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