Theorem tz6.12c 5546

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 2056	. . . 4 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ∃𝑦 𝐴𝐹𝑦)
2		nfeu1 2037	. . . . . 6 ⊢ Ⅎ𝑦∃!𝑦 𝐴𝐹𝑦
3		nfv 1528	. . . . . 6 ⊢ Ⅎ𝑦 𝐴𝐹(𝐹‘𝐴)
4	2, 3	nfim 1572	. . . . 5 ⊢ Ⅎ𝑦(∃!𝑦 𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴))
5		tz6.12-1 5543	. . . . . . . 8 ⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦)
6	5	expcom 116	. . . . . . 7 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 → (𝐹‘𝐴) = 𝑦))
7		breq2 4008	. . . . . . . 8 ⊢ ((𝐹‘𝐴) = 𝑦 → (𝐴𝐹(𝐹‘𝐴) ↔ 𝐴𝐹𝑦))
8	7	biimprd 158	. . . . . . 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 1594	. . . 4 ⊢ (∃𝑦 𝐴𝐹𝑦 → (∃!𝑦 𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴)))
12	1, 11	mpcom 36	. . 3 ⊢ (∃!𝑦 𝐴𝐹𝑦 → 𝐴𝐹(𝐹‘𝐴))
13	12, 7	syl5ibcom 155	. 2 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 → 𝐴𝐹𝑦))
14	13, 6	impbid 129	1 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1353  ∃wex 1492  ∃!weu 2026   class class class wbr 4004  ‘cfv 5217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-iota 5179  df-fv 5225

This theorem is referenced by:  fnbrfvb  5557