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Mirrors > Home > MPE Home > Th. List > op2ndb | Structured version Visualization version GIF version |
Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4940 to extract the first member, op2nda 5620 for an alternate version, and op2nd 7177 for the preferred version.) (Contributed by NM, 25-Nov-2003.) |
Ref | Expression |
---|---|
cnvsn.1 | ⊢ 𝐴 ∈ V |
cnvsn.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
op2ndb | ⊢ ∩ ∩ ∩ ◡{〈𝐴, 𝐵〉} = 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvsn.1 | . . . . . . 7 ⊢ 𝐴 ∈ V | |
2 | cnvsn.2 | . . . . . . 7 ⊢ 𝐵 ∈ V | |
3 | 1, 2 | cnvsn 5618 | . . . . . 6 ⊢ ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉} |
4 | 3 | inteqi 4479 | . . . . 5 ⊢ ∩ ◡{〈𝐴, 𝐵〉} = ∩ {〈𝐵, 𝐴〉} |
5 | opex 4932 | . . . . . 6 ⊢ 〈𝐵, 𝐴〉 ∈ V | |
6 | 5 | intsn 4513 | . . . . 5 ⊢ ∩ {〈𝐵, 𝐴〉} = 〈𝐵, 𝐴〉 |
7 | 4, 6 | eqtri 2644 | . . . 4 ⊢ ∩ ◡{〈𝐴, 𝐵〉} = 〈𝐵, 𝐴〉 |
8 | 7 | inteqi 4479 | . . 3 ⊢ ∩ ∩ ◡{〈𝐴, 𝐵〉} = ∩ 〈𝐵, 𝐴〉 |
9 | 8 | inteqi 4479 | . 2 ⊢ ∩ ∩ ∩ ◡{〈𝐴, 𝐵〉} = ∩ ∩ 〈𝐵, 𝐴〉 |
10 | 2, 1 | op1stb 4940 | . 2 ⊢ ∩ ∩ 〈𝐵, 𝐴〉 = 𝐵 |
11 | 9, 10 | eqtri 2644 | 1 ⊢ ∩ ∩ ∩ ◡{〈𝐴, 𝐵〉} = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 Vcvv 3200 {csn 4177 〈cop 4183 ∩ cint 4475 ◡ccnv 5113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-int 4476 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 |
This theorem is referenced by: 2ndval2 7186 |
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