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Mirrors > Home > MPE Home > Th. List > 2ndval2 | Structured version Visualization version GIF version |
Description: Alternate value of the function that extracts the second member of an ordered pair. Definition 5.13 (ii) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.) |
Ref | Expression |
---|---|
2ndval2 | ⊢ (𝐴 ∈ (V × V) → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elvv 5763 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | |
2 | vex 3482 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 3482 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | op2nd 8022 | . . . . 5 ⊢ (2nd ‘〈𝑥, 𝑦〉) = 𝑦 |
5 | 2, 3 | op2ndb 6249 | . . . . 5 ⊢ ∩ ∩ ∩ ◡{〈𝑥, 𝑦〉} = 𝑦 |
6 | 4, 5 | eqtr4i 2766 | . . . 4 ⊢ (2nd ‘〈𝑥, 𝑦〉) = ∩ ∩ ∩ ◡{〈𝑥, 𝑦〉} |
7 | fveq2 6907 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (2nd ‘𝐴) = (2nd ‘〈𝑥, 𝑦〉)) | |
8 | sneq 4641 | . . . . . . . 8 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → {𝐴} = {〈𝑥, 𝑦〉}) | |
9 | 8 | cnveqd 5889 | . . . . . . 7 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ◡{𝐴} = ◡{〈𝑥, 𝑦〉}) |
10 | 9 | inteqd 4956 | . . . . . 6 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∩ ◡{𝐴} = ∩ ◡{〈𝑥, 𝑦〉}) |
11 | 10 | inteqd 4956 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∩ ∩ ◡{𝐴} = ∩ ∩ ◡{〈𝑥, 𝑦〉}) |
12 | 11 | inteqd 4956 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∩ ∩ ∩ ◡{𝐴} = ∩ ∩ ∩ ◡{〈𝑥, 𝑦〉}) |
13 | 6, 7, 12 | 3eqtr4a 2801 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴}) |
14 | 13 | exlimivv 1930 | . 2 ⊢ (∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉 → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴}) |
15 | 1, 14 | sylbi 217 | 1 ⊢ (𝐴 ∈ (V × V) → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∃wex 1776 ∈ wcel 2106 Vcvv 3478 {csn 4631 〈cop 4637 ∩ cint 4951 × cxp 5687 ◡ccnv 5688 ‘cfv 6563 2nd c2nd 8012 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fv 6571 df-2nd 8014 |
This theorem is referenced by: (None) |
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