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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 5177 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | |
2 | vex 3203 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 3203 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | op2nd 7177 | . . . . 5 ⊢ (2nd ‘〈𝑥, 𝑦〉) = 𝑦 |
5 | 2, 3 | op2ndb 5619 | . . . . 5 ⊢ ∩ ∩ ∩ ◡{〈𝑥, 𝑦〉} = 𝑦 |
6 | 4, 5 | eqtr4i 2647 | . . . 4 ⊢ (2nd ‘〈𝑥, 𝑦〉) = ∩ ∩ ∩ ◡{〈𝑥, 𝑦〉} |
7 | fveq2 6191 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (2nd ‘𝐴) = (2nd ‘〈𝑥, 𝑦〉)) | |
8 | sneq 4187 | . . . . . . . 8 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → {𝐴} = {〈𝑥, 𝑦〉}) | |
9 | 8 | cnveqd 5298 | . . . . . . 7 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ◡{𝐴} = ◡{〈𝑥, 𝑦〉}) |
10 | 9 | inteqd 4480 | . . . . . 6 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∩ ◡{𝐴} = ∩ ◡{〈𝑥, 𝑦〉}) |
11 | 10 | inteqd 4480 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∩ ∩ ◡{𝐴} = ∩ ∩ ◡{〈𝑥, 𝑦〉}) |
12 | 11 | inteqd 4480 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∩ ∩ ∩ ◡{𝐴} = ∩ ∩ ∩ ◡{〈𝑥, 𝑦〉}) |
13 | 6, 7, 12 | 3eqtr4a 2682 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴}) |
14 | 13 | exlimivv 1860 | . 2 ⊢ (∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉 → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴}) |
15 | 1, 14 | sylbi 207 | 1 ⊢ (𝐴 ∈ (V × V) → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∃wex 1704 ∈ wcel 1990 Vcvv 3200 {csn 4177 〈cop 4183 ∩ cint 4475 × cxp 5112 ◡ccnv 5113 ‘cfv 5888 2nd c2nd 7167 |
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-8 1992 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-pow 4843 ax-pr 4906 ax-un 6949 |
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-sbc 3436 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-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fv 5896 df-2nd 7169 |
This theorem is referenced by: (None) |
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