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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 4402 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | |
2 | vex 2560 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 2560 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | op2nd 5774 | . . . . 5 ⊢ (2nd ‘〈𝑥, 𝑦〉) = 𝑦 |
5 | 2, 3 | op2ndb 4804 | . . . . 5 ⊢ ∩ ∩ ∩ ◡{〈𝑥, 𝑦〉} = 𝑦 |
6 | 4, 5 | eqtr4i 2063 | . . . 4 ⊢ (2nd ‘〈𝑥, 𝑦〉) = ∩ ∩ ∩ ◡{〈𝑥, 𝑦〉} |
7 | fveq2 5178 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (2nd ‘𝐴) = (2nd ‘〈𝑥, 𝑦〉)) | |
8 | sneq 3386 | . . . . . . . 8 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → {𝐴} = {〈𝑥, 𝑦〉}) | |
9 | 8 | cnveqd 4511 | . . . . . . 7 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ◡{𝐴} = ◡{〈𝑥, 𝑦〉}) |
10 | 9 | inteqd 3620 | . . . . . 6 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∩ ◡{𝐴} = ∩ ◡{〈𝑥, 𝑦〉}) |
11 | 10 | inteqd 3620 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∩ ∩ ◡{𝐴} = ∩ ∩ ◡{〈𝑥, 𝑦〉}) |
12 | 11 | inteqd 3620 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∩ ∩ ∩ ◡{𝐴} = ∩ ∩ ∩ ◡{〈𝑥, 𝑦〉}) |
13 | 6, 7, 12 | 3eqtr4a 2098 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴}) |
14 | 13 | exlimivv 1776 | . 2 ⊢ (∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉 → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴}) |
15 | 1, 14 | sylbi 114 | 1 ⊢ (𝐴 ∈ (V × V) → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ∃wex 1381 ∈ wcel 1393 Vcvv 2557 {csn 3375 〈cop 3378 ∩ cint 3615 × cxp 4343 ◡ccnv 4344 ‘cfv 4902 2nd c2nd 5766 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-iota 4867 df-fun 4904 df-fv 4910 df-2nd 5768 |
This theorem is referenced by: (None) |
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