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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 5760 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉) | |
| 2 | vex 3484 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 3 | vex 3484 | . . . . . 6 ⊢ 𝑦 ∈ V | |
| 4 | 2, 3 | op1st 8022 | . . . . 5 ⊢ (1st ‘〈𝑥, 𝑦〉) = 𝑥 |
| 5 | 2, 3 | op1stb 5476 | . . . . 5 ⊢ ∩ ∩ 〈𝑥, 𝑦〉 = 𝑥 |
| 6 | 4, 5 | eqtr4i 2768 | . . . 4 ⊢ (1st ‘〈𝑥, 𝑦〉) = ∩ ∩ 〈𝑥, 𝑦〉 |
| 7 | fveq2 6906 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (1st ‘𝐴) = (1st ‘〈𝑥, 𝑦〉)) | |
| 8 | inteq 4949 | . . . . 5 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∩ 𝐴 = ∩ 〈𝑥, 𝑦〉) | |
| 9 | 8 | inteqd 4951 | . . . 4 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → ∩ ∩ 𝐴 = ∩ ∩ 〈𝑥, 𝑦〉) |
| 10 | 6, 7, 9 | 3eqtr4a 2803 | . . 3 ⊢ (𝐴 = 〈𝑥, 𝑦〉 → (1st ‘𝐴) = ∩ ∩ 𝐴) |
| 11 | 10 | exlimivv 1932 | . 2 ⊢ (∃𝑥∃𝑦 𝐴 = 〈𝑥, 𝑦〉 → (1st ‘𝐴) = ∩ ∩ 𝐴) |
| 12 | 1, 11 | sylbi 217 | 1 ⊢ (𝐴 ∈ (V × V) → (1st ‘𝐴) = ∩ ∩ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 〈cop 4632 ∩ cint 4946 × cxp 5683 ‘cfv 6561 1st c1st 8012 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-1st 8014 |
| This theorem is referenced by: 1stdm 8065 |
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