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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 5619 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | |
2 | vex 3495 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 3495 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | op2nd 7687 | . . . . 5 ⊢ (2nd ‘〈𝑥, 𝑦〉) = 𝑦 |
5 | 2, 3 | op2ndb 6077 | . . . . 5 ⊢ ∩ ∩ ∩ ◡{〈𝑥, 𝑦〉} = 𝑦 |
6 | 4, 5 | eqtr4i 2844 | . . . 4 ⊢ (2nd ‘〈𝑥, 𝑦〉) = ∩ ∩ ∩ ◡{〈𝑥, 𝑦〉} |
7 | fveq2 6663 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (2nd ‘𝐴) = (2nd ‘〈𝑥, 𝑦〉)) | |
8 | sneq 4567 | . . . . . . . 8 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → {𝐴} = {〈𝑥, 𝑦〉}) | |
9 | 8 | cnveqd 5739 | . . . . . . 7 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ◡{𝐴} = ◡{〈𝑥, 𝑦〉}) |
10 | 9 | inteqd 4872 | . . . . . 6 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∩ ◡{𝐴} = ∩ ◡{〈𝑥, 𝑦〉}) |
11 | 10 | inteqd 4872 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∩ ∩ ◡{𝐴} = ∩ ∩ ◡{〈𝑥, 𝑦〉}) |
12 | 11 | inteqd 4872 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∩ ∩ ∩ ◡{𝐴} = ∩ ∩ ∩ ◡{〈𝑥, 𝑦〉}) |
13 | 6, 7, 12 | 3eqtr4a 2879 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴}) |
14 | 13 | exlimivv 1924 | . 2 ⊢ (∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉 → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴}) |
15 | 1, 14 | sylbi 218 | 1 ⊢ (𝐴 ∈ (V × V) → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∃wex 1771 ∈ wcel 2105 Vcvv 3492 {csn 4557 〈cop 4563 ∩ cint 4867 × cxp 5546 ◡ccnv 5547 ‘cfv 6348 2nd c2nd 7677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-int 4868 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fv 6356 df-2nd 7679 |
This theorem is referenced by: (None) |
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