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Theorem tz6.12-1 5143
 Description: Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
Assertion
Ref Expression
tz6.12-1 ((A𝐹y ∃!y A𝐹y) → (𝐹A) = y)
Distinct variable groups:   y,𝐹   y,A

Proof of Theorem tz6.12-1
StepHypRef Expression
1 df-fv 4853 . 2 (𝐹A) = (℩yA𝐹y)
2 iota1 4824 . . 3 (∃!y A𝐹y → (A𝐹y ↔ (℩yA𝐹y) = y))
32biimpac 282 . 2 ((A𝐹y ∃!y A𝐹y) → (℩yA𝐹y) = y)
41, 3syl5eq 2081 1 ((A𝐹y ∃!y A𝐹y) → (𝐹A) = y)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242  ∃!weu 1897   class class class wbr 3755  ℩cio 4808  ‘cfv 4845 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-sn 3373  df-pr 3374  df-uni 3572  df-iota 4810  df-fv 4853 This theorem is referenced by:  tz6.12  5144  tz6.12c  5146  funbrfv  5155
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