![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > xp1st | GIF version |
Description: Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
xp1st | ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (1st ‘𝐴) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp 4494 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑏∃𝑐(𝐴 = 〈𝑏, 𝑐〉 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶))) | |
2 | vex 2644 | . . . . . . 7 ⊢ 𝑏 ∈ V | |
3 | vex 2644 | . . . . . . 7 ⊢ 𝑐 ∈ V | |
4 | 2, 3 | op1std 5977 | . . . . . 6 ⊢ (𝐴 = 〈𝑏, 𝑐〉 → (1st ‘𝐴) = 𝑏) |
5 | 4 | eleq1d 2168 | . . . . 5 ⊢ (𝐴 = 〈𝑏, 𝑐〉 → ((1st ‘𝐴) ∈ 𝐵 ↔ 𝑏 ∈ 𝐵)) |
6 | 5 | biimpar 293 | . . . 4 ⊢ ((𝐴 = 〈𝑏, 𝑐〉 ∧ 𝑏 ∈ 𝐵) → (1st ‘𝐴) ∈ 𝐵) |
7 | 6 | adantrr 466 | . . 3 ⊢ ((𝐴 = 〈𝑏, 𝑐〉 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (1st ‘𝐴) ∈ 𝐵) |
8 | 7 | exlimivv 1835 | . 2 ⊢ (∃𝑏∃𝑐(𝐴 = 〈𝑏, 𝑐〉 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (1st ‘𝐴) ∈ 𝐵) |
9 | 1, 8 | sylbi 120 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → (1st ‘𝐴) ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1299 ∃wex 1436 ∈ wcel 1448 〈cop 3477 × cxp 4475 ‘cfv 5059 1st c1st 5967 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-sbc 2863 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-iota 5024 df-fun 5061 df-fv 5067 df-1st 5969 |
This theorem is referenced by: disjxp1 6063 xpf1o 6667 xpmapenlem 6672 djuf1olem 6853 eldju1st 6871 dfplpq2 7063 dfmpq2 7064 enqbreq2 7066 enqdc1 7071 mulpipq2 7080 preqlu 7181 elnp1st2nd 7185 cauappcvgprlemladd 7367 elreal2 7518 cnref1o 9290 frecuzrdgrrn 10022 frec2uzrdg 10023 frecuzrdgrcl 10024 frecuzrdgsuc 10028 frecuzrdgrclt 10029 frecuzrdgg 10030 frecuzrdgsuctlem 10037 seq3val 10072 seqvalcd 10073 fsum2dlemstep 11042 fisumcom2 11046 eucalgval 11528 eucalginv 11530 eucalglt 11531 eucalg 11533 sqpweven 11645 2sqpwodd 11646 tx2cn 12220 txdis 12227 |
Copyright terms: Public domain | W3C validator |