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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 5743 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = ⟨𝑥, 𝑦⟩) | |
2 | vex 3472 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | vex 3472 | . . . . . 6 ⊢ 𝑦 ∈ V | |
4 | 2, 3 | op1st 7979 | . . . . 5 ⊢ (1st ‘⟨𝑥, 𝑦⟩) = 𝑥 |
5 | 2, 3 | op1stb 5464 | . . . . 5 ⊢ ∩ ∩ ⟨𝑥, 𝑦⟩ = 𝑥 |
6 | 4, 5 | eqtr4i 2757 | . . . 4 ⊢ (1st ‘⟨𝑥, 𝑦⟩) = ∩ ∩ ⟨𝑥, 𝑦⟩ |
7 | fveq2 6884 | . . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1st ‘𝐴) = (1st ‘⟨𝑥, 𝑦⟩)) | |
8 | inteq 4946 | . . . . 5 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ 𝐴 = ∩ ⟨𝑥, 𝑦⟩) | |
9 | 8 | inteqd 4948 | . . . 4 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∩ ∩ 𝐴 = ∩ ∩ ⟨𝑥, 𝑦⟩) |
10 | 6, 7, 9 | 3eqtr4a 2792 | . . 3 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → (1st ‘𝐴) = ∩ ∩ 𝐴) |
11 | 10 | exlimivv 1927 | . 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 1533 ∃wex 1773 ∈ wcel 2098 Vcvv 3468 ⟨cop 4629 ∩ cint 4943 × cxp 5667 ‘cfv 6536 1st c1st 7969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6488 df-fun 6538 df-fv 6544 df-1st 7971 |
This theorem is referenced by: 1stdm 8022 |
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