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Mirrors > Home > ILE Home > Th. List > tz6.12-1 | GIF version |
Description: Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) |
Ref | Expression |
---|---|
tz6.12-1 | ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fv 5196 | . 2 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦) | |
2 | iota1 5167 | . . 3 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 ↔ (℩𝑦𝐴𝐹𝑦) = 𝑦)) | |
3 | 2 | biimpac 296 | . 2 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (℩𝑦𝐴𝐹𝑦) = 𝑦) |
4 | 1, 3 | syl5eq 2211 | 1 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1343 ∃!weu 2014 class class class wbr 3982 ℩cio 5151 ‘cfv 5188 |
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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-ext 2147 |
This theorem depends on definitions: df-bi 116 df-tru 1346 df-nf 1449 df-sb 1751 df-eu 2017 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-rex 2450 df-v 2728 df-sbc 2952 df-un 3120 df-sn 3582 df-pr 3583 df-uni 3790 df-iota 5153 df-fv 5196 |
This theorem is referenced by: tz6.12 5514 tz6.12c 5516 funbrfv 5525 |
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