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Mirrors > Home > MPE Home > Th. List > tz6.12c | Structured version Visualization version GIF version |
Description: Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by SN, 23-Dec-2024.) |
Ref | Expression |
---|---|
tz6.12c | ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fv 6509 | . . 3 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦) | |
2 | 1 | eqeq1i 2742 | . 2 ⊢ ((𝐹‘𝐴) = 𝑦 ↔ (℩𝑦𝐴𝐹𝑦) = 𝑦) |
3 | iota1 6478 | . 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 ↔ (℩𝑦𝐴𝐹𝑦) = 𝑦)) | |
4 | 2, 3 | bitr4id 290 | 1 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∃!weu 2567 class class class wbr 5110 ℩cio 6451 ‘cfv 6501 |
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 2708 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3450 df-un 3920 df-in 3922 df-ss 3932 df-sn 4592 df-pr 4594 df-uni 4871 df-iota 6453 df-fv 6509 |
This theorem is referenced by: tz6.12-1 6870 tz6.12i 6875 fnbrfvb 6900 |
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