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Mirrors > Home > MPE Home > Th. List > tz6.12 | Structured version Visualization version GIF version |
Description: Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 10-Jul-1994.) |
Ref | Expression |
---|---|
tz6.12 | ⊢ ((〈𝐴, 𝑦〉 ∈ 𝐹 ∧ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) → (𝐹‘𝐴) = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5149 | . 2 ⊢ (𝐴𝐹𝑦 ↔ 〈𝐴, 𝑦〉 ∈ 𝐹) | |
2 | 1 | eubii 2583 | . 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 ↔ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) |
3 | tz6.12-1 6930 | . 2 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦) | |
4 | 1, 2, 3 | syl2anbr 599 | 1 ⊢ ((〈𝐴, 𝑦〉 ∈ 𝐹 ∧ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) → (𝐹‘𝐴) = 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃!weu 2566 〈cop 4637 class class class wbr 5148 ‘cfv 6563 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-ss 3980 df-sn 4632 df-pr 4634 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 |
This theorem is referenced by: tz6.12f 6933 dfac5lem5 10165 tz6.12-afv 47123 |
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