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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 6358 | . 2 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦) | |
2 | iota1 6327 | . . 3 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 ↔ (℩𝑦𝐴𝐹𝑦) = 𝑦)) | |
3 | 2 | biimpac 481 | . 2 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (℩𝑦𝐴𝐹𝑦) = 𝑦) |
4 | 1, 3 | syl5eq 2868 | 1 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∃!weu 2649 class class class wbr 5059 ℩cio 6307 ‘cfv 6350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-v 3497 df-sbc 3773 df-un 3941 df-sn 4562 df-pr 4564 df-uni 4833 df-iota 6309 df-fv 6358 |
This theorem is referenced by: tz6.12 6688 tz6.12c 6690 funbrfv 6711 setrec2lem2 44790 |
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