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Mirrors > Home > ILE Home > Th. List > elxp6 | GIF version |
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5085. (Contributed by NM, 9-Oct-2004.) |
Ref | Expression |
---|---|
elxp6 | ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2732 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ V) | |
2 | opexg 4200 | . . . 4 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶) → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ V) | |
3 | 2 | adantl 275 | . . 3 ⊢ ((𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ V) |
4 | eleq1 2227 | . . . 4 ⊢ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 → (𝐴 ∈ V ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ V)) | |
5 | 4 | adantr 274 | . . 3 ⊢ ((𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → (𝐴 ∈ V ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ V)) |
6 | 3, 5 | mpbird 166 | . 2 ⊢ ((𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → 𝐴 ∈ V) |
7 | elxp4 5085 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = 〈∪ dom {𝐴}, ∪ ran {𝐴}〉 ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))) | |
8 | 1stvalg 6102 | . . . . . 6 ⊢ (𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴}) | |
9 | 2ndvalg 6103 | . . . . . 6 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴}) | |
10 | 8, 9 | opeq12d 3760 | . . . . 5 ⊢ (𝐴 ∈ V → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈∪ dom {𝐴}, ∪ ran {𝐴}〉) |
11 | 10 | eqeq2d 2176 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ↔ 𝐴 = 〈∪ dom {𝐴}, ∪ ran {𝐴}〉)) |
12 | 8 | eleq1d 2233 | . . . . 5 ⊢ (𝐴 ∈ V → ((1st ‘𝐴) ∈ 𝐵 ↔ ∪ dom {𝐴} ∈ 𝐵)) |
13 | 9 | eleq1d 2233 | . . . . 5 ⊢ (𝐴 ∈ V → ((2nd ‘𝐴) ∈ 𝐶 ↔ ∪ ran {𝐴} ∈ 𝐶)) |
14 | 12, 13 | anbi12d 465 | . . . 4 ⊢ (𝐴 ∈ V → (((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶) ↔ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))) |
15 | 11, 14 | anbi12d 465 | . . 3 ⊢ (𝐴 ∈ V → ((𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) ↔ (𝐴 = 〈∪ dom {𝐴}, ∪ ran {𝐴}〉 ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))) |
16 | 7, 15 | bitr4id 198 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))) |
17 | 1, 6, 16 | pm5.21nii 694 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1342 ∈ wcel 2135 Vcvv 2721 {csn 3570 〈cop 3573 ∪ cuni 3783 × cxp 4596 dom cdm 4598 ran crn 4599 ‘cfv 5182 1st c1st 6098 2nd c2nd 6099 |
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 df-2nd 6101 |
This theorem is referenced by: elxp7 6130 oprssdmm 6131 eqopi 6132 1st2nd2 6135 eldju2ndl 7028 eldju2ndr 7029 qredeu 12008 qnumdencl 12096 tx1cn 12810 tx2cn 12811 psmetxrge0 12873 xmetxpbl 13049 |
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