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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 5535 to extract the first member, op2ndb 5537 for an alternate version, and op2nd 7068 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 5534 | . . 3 ⊢ ran {〈𝐴, 𝐵〉} = {𝐵} |
3 | 2 | unieqi 4381 | . 2 ⊢ ∪ ran {〈𝐴, 𝐵〉} = ∪ {𝐵} |
4 | cnvsn.2 | . . 3 ⊢ 𝐵 ∈ V | |
5 | 4 | unisn 4387 | . 2 ⊢ ∪ {𝐵} = 𝐵 |
6 | 3, 5 | eqtri 2632 | 1 ⊢ ∪ ran {〈𝐴, 𝐵〉} = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 〈cop 4131 ∪ cuni 4372 ran crn 5039 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
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-rex 2902 df-rab 2905 df-v 3175 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-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: elxp4 7003 elxp5 7004 op2nd 7068 fo2nd 7080 f2ndres 7082 ixpsnf1o 7834 xpassen 7939 xpdom2 7940 |
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