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Mirrors > Home > ILE Home > Th. List > 2ndval2 | 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 4596 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | |
2 | vex 2684 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 2684 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | op2nd 6038 | . . . . 5 ⊢ (2nd ‘〈𝑥, 𝑦〉) = 𝑦 |
5 | 2, 3 | op2ndb 5017 | . . . . 5 ⊢ ∩ ∩ ∩ ◡{〈𝑥, 𝑦〉} = 𝑦 |
6 | 4, 5 | eqtr4i 2161 | . . . 4 ⊢ (2nd ‘〈𝑥, 𝑦〉) = ∩ ∩ ∩ ◡{〈𝑥, 𝑦〉} |
7 | fveq2 5414 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (2nd ‘𝐴) = (2nd ‘〈𝑥, 𝑦〉)) | |
8 | sneq 3533 | . . . . . . . 8 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → {𝐴} = {〈𝑥, 𝑦〉}) | |
9 | 8 | cnveqd 4710 | . . . . . . 7 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ◡{𝐴} = ◡{〈𝑥, 𝑦〉}) |
10 | 9 | inteqd 3771 | . . . . . 6 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∩ ◡{𝐴} = ∩ ◡{〈𝑥, 𝑦〉}) |
11 | 10 | inteqd 3771 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∩ ∩ ◡{𝐴} = ∩ ∩ ◡{〈𝑥, 𝑦〉}) |
12 | 11 | inteqd 3771 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∩ ∩ ∩ ◡{𝐴} = ∩ ∩ ∩ ◡{〈𝑥, 𝑦〉}) |
13 | 6, 7, 12 | 3eqtr4a 2196 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴}) |
14 | 13 | exlimivv 1868 | . 2 ⊢ (∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉 → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴}) |
15 | 1, 14 | sylbi 120 | 1 ⊢ (𝐴 ∈ (V × V) → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∃wex 1468 ∈ wcel 1480 Vcvv 2681 {csn 3522 〈cop 3525 ∩ cint 3766 × cxp 4532 ◡ccnv 4533 ‘cfv 5118 2nd c2nd 6030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-iota 5083 df-fun 5120 df-fv 5126 df-2nd 6032 |
This theorem is referenced by: (None) |
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