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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 6178 to extract the first member, op2ndb 6180 for an alternate version, and op2nd 7931 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 6177 | . . 3 ⊢ ran {⟨𝐴, 𝐵⟩} = {𝐵} |
3 | 2 | unieqi 4879 | . 2 ⊢ ∪ ran {⟨𝐴, 𝐵⟩} = ∪ {𝐵} |
4 | cnvsn.2 | . . 3 ⊢ 𝐵 ∈ V | |
5 | 4 | unisn 4888 | . 2 ⊢ ∪ {𝐵} = 𝐵 |
6 | 3, 5 | eqtri 2765 | 1 ⊢ ∪ ran {⟨𝐴, 𝐵⟩} = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 Vcvv 3446 {csn 4587 ⟨cop 4593 ∪ cuni 4866 ran crn 5635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-xp 5640 df-rel 5641 df-cnv 5642 df-dm 5644 df-rn 5645 |
This theorem is referenced by: elxp4 7860 elxp5 7861 op2nd 7931 fo2nd 7943 f2ndres 7947 ixpsnf1o 8877 xpassen 9011 xpdom2 9012 |
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