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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 5134. (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 2763 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ V) | |
2 | opexg 4246 | . . . 4 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶) → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ V) | |
3 | 2 | adantl 277 | . . 3 ⊢ ((𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ V) |
4 | eleq1 2252 | . . . 4 ⊢ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 → (𝐴 ∈ V ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ V)) | |
5 | 4 | adantr 276 | . . 3 ⊢ ((𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → (𝐴 ∈ V ↔ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ V)) |
6 | 3, 5 | mpbird 167 | . 2 ⊢ ((𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → 𝐴 ∈ V) |
7 | elxp4 5134 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = 〈∪ dom {𝐴}, ∪ ran {𝐴}〉 ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))) | |
8 | 1stvalg 6168 | . . . . . 6 ⊢ (𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴}) | |
9 | 2ndvalg 6169 | . . . . . 6 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴}) | |
10 | 8, 9 | opeq12d 3801 | . . . . 5 ⊢ (𝐴 ∈ V → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈∪ dom {𝐴}, ∪ ran {𝐴}〉) |
11 | 10 | eqeq2d 2201 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ↔ 𝐴 = 〈∪ dom {𝐴}, ∪ ran {𝐴}〉)) |
12 | 8 | eleq1d 2258 | . . . . 5 ⊢ (𝐴 ∈ V → ((1st ‘𝐴) ∈ 𝐵 ↔ ∪ dom {𝐴} ∈ 𝐵)) |
13 | 9 | eleq1d 2258 | . . . . 5 ⊢ (𝐴 ∈ V → ((2nd ‘𝐴) ∈ 𝐶 ↔ ∪ ran {𝐴} ∈ 𝐶)) |
14 | 12, 13 | anbi12d 473 | . . . 4 ⊢ (𝐴 ∈ V → (((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶) ↔ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))) |
15 | 11, 14 | anbi12d 473 | . . 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 199 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))) |
17 | 1, 6, 16 | pm5.21nii 705 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2160 Vcvv 2752 {csn 3607 〈cop 3610 ∪ cuni 3824 × cxp 4642 dom cdm 4644 ran crn 4645 ‘cfv 5235 1st c1st 6164 2nd c2nd 6165 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-iota 5196 df-fun 5237 df-fv 5243 df-1st 6166 df-2nd 6167 |
This theorem is referenced by: elxp7 6196 oprssdmm 6197 eqopi 6198 1st2nd2 6201 eldju2ndl 7102 eldju2ndr 7103 aptap 8638 qredeu 12132 qnumdencl 12222 tx1cn 14246 tx2cn 14247 psmetxrge0 14309 xmetxpbl 14485 |
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