Theorem tz6.12c 5459

Description: Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)

Assertion

Ref	Expression
tz6.12c	⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))

Distinct variable groups:   𝑦,𝐹   𝑦,𝐴

Proof of Theorem tz6.12c

Step	Hyp	Ref	Expression
1		euex 2030	. . . 4 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃𝑦 𝐴𝐹𝑦)
2		nfeu1 2011	. . . . . 6 ⊢ Ⅎ𝑦∃!𝑦 𝐴𝐹𝑦
3		nfv 1509	. . . . . 6 ⊢ Ⅎ𝑦 𝐴𝐹(𝐹‘𝐴)
4	2, 3	nfim 1552	. . . . 5 ⊢ Ⅎ𝑦(∃!𝑦 𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴))
5		tz6.12-1 5456	. . . . . . . 8 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦)
6	5	expcom 115	. . . . . . 7 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → (𝐹‘𝐴) = 𝑦))
7		breq2 3941	. . . . . . . 8 ⊢ ((𝐹‘𝐴) = 𝑦 → (𝐴𝐹(𝐹‘𝐴) ↔ 𝐴𝐹𝑦))
8	7	biimprd 157	. . . . . . 7 ⊢ ((𝐹‘𝐴) = 𝑦 → (𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴)))
9	6, 8	syli 37	. . . . . 6 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴)))
10	9	com12 30	. . . . 5 ⊢ (𝐴𝐹𝑦 → (∃!𝑦 𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴)))
11	4, 10	exlimi 1574	. . . 4 ⊢ (∃𝑦 𝐴𝐹𝑦 → (∃!𝑦 𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴)))
12	1, 11	mpcom 36	. . 3 ⊢ (∃!𝑦 𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴))
13	12, 7	syl5ibcom 154	. 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 → 𝐴𝐹𝑦))
14	13, 6	impbid 128	1 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1332  ∃wex 1469  ∃!weu 2000   class class class wbr 3937  ‘cfv 5131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-iota 5096  df-fv 5139

This theorem is referenced by:  fnbrfvb  5470