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Mirrors > Home > MPE Home > Th. List > 1stval2 | Structured version Visualization version GIF version |
Description: Alternate value of the function that extracts the first member of an ordered pair. Definition 5.13 (i) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.) |
Ref | Expression |
---|---|
1stval2 | ⊢ (𝐴 ∈ (V × V) → (1st ‘𝐴) = ∩ ∩ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elvv 5661 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | |
2 | vex 3436 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 3436 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | op1st 7839 | . . . . 5 ⊢ (1st ‘〈𝑥, 𝑦〉) = 𝑥 |
5 | 2, 3 | op1stb 5386 | . . . . 5 ⊢ ∩ ∩ 〈𝑥, 𝑦〉 = 𝑥 |
6 | 4, 5 | eqtr4i 2769 | . . . 4 ⊢ (1st ‘〈𝑥, 𝑦〉) = ∩ ∩ 〈𝑥, 𝑦〉 |
7 | fveq2 6774 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (1st ‘𝐴) = (1st ‘〈𝑥, 𝑦〉)) | |
8 | inteq 4882 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∩ 𝐴 = ∩ 〈𝑥, 𝑦〉) | |
9 | 8 | inteqd 4884 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∩ ∩ 𝐴 = ∩ ∩ 〈𝑥, 𝑦〉) |
10 | 6, 7, 9 | 3eqtr4a 2804 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (1st ‘𝐴) = ∩ ∩ 𝐴) |
11 | 10 | exlimivv 1935 | . 2 ⊢ (∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉 → (1st ‘𝐴) = ∩ ∩ 𝐴) |
12 | 1, 11 | sylbi 216 | 1 ⊢ (𝐴 ∈ (V × V) → (1st ‘𝐴) = ∩ ∩ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∃wex 1782 ∈ wcel 2106 Vcvv 3432 〈cop 4567 ∩ cint 4879 × cxp 5587 ‘cfv 6433 1st c1st 7829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-iota 6391 df-fun 6435 df-fv 6441 df-1st 7831 |
This theorem is referenced by: 1stdm 7881 |
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