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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 5098. (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 2741 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ V) | |
2 | opexg 4213 | . . . 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 2233 | . . . 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 5098 | . . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = 〈∪ dom {𝐴}, ∪ ran {𝐴}〉 ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))) | |
8 | 1stvalg 6121 | . . . . . 6 ⊢ (𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴}) | |
9 | 2ndvalg 6122 | . . . . . 6 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴}) | |
10 | 8, 9 | opeq12d 3773 | . . . . 5 ⊢ (𝐴 ∈ V → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 = 〈∪ dom {𝐴}, ∪ ran {𝐴}〉) |
11 | 10 | eqeq2d 2182 | . . . 4 ⊢ (𝐴 ∈ V → (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ↔ 𝐴 = 〈∪ dom {𝐴}, ∪ ran {𝐴}〉)) |
12 | 8 | eleq1d 2239 | . . . . 5 ⊢ (𝐴 ∈ V → ((1st ‘𝐴) ∈ 𝐵 ↔ ∪ dom {𝐴} ∈ 𝐵)) |
13 | 9 | eleq1d 2239 | . . . . 5 ⊢ (𝐴 ∈ V → ((2nd ‘𝐴) ∈ 𝐶 ↔ ∪ ran {𝐴} ∈ 𝐶)) |
14 | 12, 13 | anbi12d 470 | . . . 4 ⊢ (𝐴 ∈ V → (((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶) ↔ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))) |
15 | 11, 14 | anbi12d 470 | . . 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 699 | 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 1348 ∈ wcel 2141 Vcvv 2730 {csn 3583 〈cop 3586 ∪ cuni 3796 × cxp 4609 dom cdm 4611 ran crn 4612 ‘cfv 5198 1st c1st 6117 2nd c2nd 6118 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-iota 5160 df-fun 5200 df-fv 5206 df-1st 6119 df-2nd 6120 |
This theorem is referenced by: elxp7 6149 oprssdmm 6150 eqopi 6151 1st2nd2 6154 eldju2ndl 7049 eldju2ndr 7050 qredeu 12051 qnumdencl 12141 tx1cn 13063 tx2cn 13064 psmetxrge0 13126 xmetxpbl 13302 |
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