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Mirrors > Home > MPE Home > Th. List > tz6.12-1 | Structured version Visualization version 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 5812 | . 2 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦) | |
2 | iota1 5782 | . . 3 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 ↔ (℩𝑦𝐴𝐹𝑦) = 𝑦)) | |
3 | 2 | biimpac 502 | . 2 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (℩𝑦𝐴𝐹𝑦) = 𝑦) |
4 | 1, 3 | syl5eq 2656 | 1 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∃!weu 2458 class class class wbr 4583 ℩cio 5766 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-v 3175 df-sbc 3403 df-un 3545 df-sn 4126 df-pr 4128 df-uni 4373 df-iota 5768 df-fv 5812 |
This theorem is referenced by: tz6.12 6121 tz6.12c 6123 funbrfv 6144 setrec2lem2 42240 |
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