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Mirrors > Home > ILE Home > Th. List > tz6.12 | GIF version |
Description: Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 10-Jul-1994.) |
Ref | Expression |
---|---|
tz6.12 | ⊢ ((〈𝐴, 𝑦〉 ∈ 𝐹 ∧ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) → (𝐹‘𝐴) = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 3786 | . 2 ⊢ (𝐴𝐹𝑦 ↔ 〈𝐴, 𝑦〉 ∈ 𝐹) | |
2 | 1 | eubii 1950 | . 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 ↔ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) |
3 | tz6.12-1 5221 | . 2 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦) | |
4 | 1, 2, 3 | syl2anbr 286 | 1 ⊢ ((〈𝐴, 𝑦〉 ∈ 𝐹 ∧ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) → (𝐹‘𝐴) = 𝑦) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 ∃!weu 1941 〈cop 3401 class class class wbr 3785 ‘cfv 4922 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-sn 3404 df-pr 3405 df-uni 3602 df-br 3786 df-iota 4887 df-fv 4930 |
This theorem is referenced by: tz6.12f 5223 |
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