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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 5750 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | |
2 | vex 3477 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 3477 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | op1st 7987 | . . . . 5 ⊢ (1st ‘〈𝑥, 𝑦〉) = 𝑥 |
5 | 2, 3 | op1stb 5471 | . . . . 5 ⊢ ∩ ∩ 〈𝑥, 𝑦〉 = 𝑥 |
6 | 4, 5 | eqtr4i 2762 | . . . 4 ⊢ (1st ‘〈𝑥, 𝑦〉) = ∩ ∩ 〈𝑥, 𝑦〉 |
7 | fveq2 6891 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (1st ‘𝐴) = (1st ‘〈𝑥, 𝑦〉)) | |
8 | inteq 4953 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∩ 𝐴 = ∩ 〈𝑥, 𝑦〉) | |
9 | 8 | inteqd 4955 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∩ ∩ 𝐴 = ∩ ∩ 〈𝑥, 𝑦〉) |
10 | 6, 7, 9 | 3eqtr4a 2797 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (1st ‘𝐴) = ∩ ∩ 𝐴) |
11 | 10 | exlimivv 1934 | . 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 1540 ∃wex 1780 ∈ wcel 2105 Vcvv 3473 〈cop 4634 ∩ cint 4950 × cxp 5674 ‘cfv 6543 1st c1st 7977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fv 6551 df-1st 7979 |
This theorem is referenced by: 1stdm 8030 |
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