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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.) (Proof shortened by SN, 23-Dec-2024.) |
Ref | Expression |
---|---|
tz6.12-1 | ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tz6.12c 6910 | . 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦)) | |
2 | 1 | biimparc 481 | 1 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∃!weu 2563 class class class wbr 5147 ‘cfv 6540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-un 3952 df-in 3954 df-ss 3964 df-sn 4628 df-pr 4630 df-uni 4908 df-iota 6492 df-fv 6548 |
This theorem is referenced by: tz6.12 6913 tz6.12cOLD 6915 funbrfv 6939 setrec2lem2 47641 |
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