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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 5707 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩) | |
2 | vex 3450 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 3450 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | op1st 7930 | . . . . 5 ⊢ (1st ‘⟨𝑥, 𝑦⟩) = 𝑥 |
5 | 2, 3 | op1stb 5429 | . . . . 5 ⊢ ∩ ∩ ⟨𝑥, 𝑦⟩ = 𝑥 |
6 | 4, 5 | eqtr4i 2768 | . . . 4 ⊢ (1st ‘⟨𝑥, 𝑦⟩) = ∩ ∩ ⟨𝑥, 𝑦⟩ |
7 | fveq2 6843 | . . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1st ‘𝐴) = (1st ‘⟨𝑥, 𝑦⟩)) | |
8 | inteq 4911 | . . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ 𝐴 = ∩ ⟨𝑥, 𝑦⟩) | |
9 | 8 | inteqd 4913 | . . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ 𝐴 = ∩ ∩ ⟨𝑥, 𝑦⟩) |
10 | 6, 7, 9 | 3eqtr4a 2803 | . . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1st ‘𝐴) = ∩ ∩ 𝐴) |
11 | 10 | exlimivv 1936 | . 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 1542 ∃wex 1782 ∈ wcel 2107 Vcvv 3446 ⟨cop 4593 ∩ cint 4908 × cxp 5632 ‘cfv 6497 1st c1st 7920 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fv 6505 df-1st 7922 |
This theorem is referenced by: 1stdm 7973 |
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