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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 4615 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑏∃𝑐(𝐴 = 〈𝑏, 𝑐〉 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶))) | |
2 | vex 2724 | . . . . . . 7 ⊢ 𝑏 ∈ V | |
3 | vex 2724 | . . . . . . 7 ⊢ 𝑐 ∈ V | |
4 | 2, 3 | op1std 6108 | . . . . . 6 ⊢ (𝐴 = 〈𝑏, 𝑐〉 → (1st ‘𝐴) = 𝑏) |
5 | 4 | eleq1d 2233 | . . . . 5 ⊢ (𝐴 = 〈𝑏, 𝑐〉 → ((1st ‘𝐴) ∈ 𝐵 ↔ 𝑏 ∈ 𝐵)) |
6 | 5 | biimpar 295 | . . . 4 ⊢ ((𝐴 = 〈𝑏, 𝑐〉 ∧ 𝑏 ∈ 𝐵) → (1st ‘𝐴) ∈ 𝐵) |
7 | 6 | adantrr 471 | . . 3 ⊢ ((𝐴 = 〈𝑏, 𝑐〉 ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (1st ‘𝐴) ∈ 𝐵) |
8 | 7 | exlimivv 1883 | . 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 1342 ∃wex 1479 ∈ wcel 2135 〈cop 3573 × cxp 4596 ‘cfv 5182 1st c1st 6098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-iota 5147 df-fun 5184 df-fv 5190 df-1st 6100 |
This theorem is referenced by: disjxp1 6195 xpf1o 6801 xpmapenlem 6806 djuf1olem 7009 eldju1st 7027 dfplpq2 7286 dfmpq2 7287 enqbreq2 7289 enqdc1 7294 mulpipq2 7303 preqlu 7404 elnp1st2nd 7408 cauappcvgprlemladd 7590 elreal2 7762 cnref1o 9579 frecuzrdgrrn 10333 frec2uzrdg 10334 frecuzrdgrcl 10335 frecuzrdgsuc 10339 frecuzrdgrclt 10340 frecuzrdgg 10341 frecuzrdgsuctlem 10348 seq3val 10383 seqvalcd 10384 fsum2dlemstep 11361 fisumcom2 11365 fprod2dlemstep 11549 fprodcom2fi 11553 eucalgval 11965 eucalginv 11967 eucalglt 11968 eucalg 11970 sqpweven 12084 2sqpwodd 12085 ctiunctlemudc 12307 tx2cn 12811 txdis 12818 txhmeo 12860 xmetxp 13048 xmetxpbl 13049 xmettxlem 13050 xmettx 13051 |
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