Theorem tz6.12i

Description: Corollary of Theorem 6.12(2) of TakeutiZaring p. 27. (Contributed by Mario Carneiro, 17-Nov-2014)

	Ref	Expression
Assertion	tz6.12i	⊢ ( 𝐵 ≠ ∅ → ( ( 𝐹 ‘ 𝐴 ) = 𝐵 → 𝐴 𝐹 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		fvex	⊢ ( 𝐹 ‘ 𝐴 ) ∈ V
2		neeq1	⊢ ( ( 𝐹 ‘ 𝐴 ) = 𝑦 → ( ( 𝐹 ‘ 𝐴 ) ≠ ∅ ↔ 𝑦 ≠ ∅ ) )
3		tz6.12-2	⊢ ( ¬ ∃! 𝑦 𝐴 𝐹 𝑦 → ( 𝐹 ‘ 𝐴 ) = ∅ )
4	3	necon1ai	⊢ ( ( 𝐹 ‘ 𝐴 ) ≠ ∅ → ∃! 𝑦 𝐴 𝐹 𝑦 )
5		tz6.12c	⊢ ( ∃! 𝑦 𝐴 𝐹 𝑦 → ( ( 𝐹 ‘ 𝐴 ) = 𝑦 ↔ 𝐴 𝐹 𝑦 ) )
6	4 5	syl	⊢ ( ( 𝐹 ‘ 𝐴 ) ≠ ∅ → ( ( 𝐹 ‘ 𝐴 ) = 𝑦 ↔ 𝐴 𝐹 𝑦 ) )
7	6	biimpcd	⊢ ( ( 𝐹 ‘ 𝐴 ) = 𝑦 → ( ( 𝐹 ‘ 𝐴 ) ≠ ∅ → 𝐴 𝐹 𝑦 ) )
8	2 7	sylbird	⊢ ( ( 𝐹 ‘ 𝐴 ) = 𝑦 → ( 𝑦 ≠ ∅ → 𝐴 𝐹 𝑦 ) )
9	8	eqcoms	⊢ ( 𝑦 = ( 𝐹 ‘ 𝐴 ) → ( 𝑦 ≠ ∅ → 𝐴 𝐹 𝑦 ) )
10		neeq1	⊢ ( 𝑦 = ( 𝐹 ‘ 𝐴 ) → ( 𝑦 ≠ ∅ ↔ ( 𝐹 ‘ 𝐴 ) ≠ ∅ ) )
11		breq2	⊢ ( 𝑦 = ( 𝐹 ‘ 𝐴 ) → ( 𝐴 𝐹 𝑦 ↔ 𝐴 𝐹 ( 𝐹 ‘ 𝐴 ) ) )
12	9 10 11	3imtr3d	⊢ ( 𝑦 = ( 𝐹 ‘ 𝐴 ) → ( ( 𝐹 ‘ 𝐴 ) ≠ ∅ → 𝐴 𝐹 ( 𝐹 ‘ 𝐴 ) ) )
13	1 12	vtocle	⊢ ( ( 𝐹 ‘ 𝐴 ) ≠ ∅ → 𝐴 𝐹 ( 𝐹 ‘ 𝐴 ) )
14	13	a1i	⊢ ( ( 𝐹 ‘ 𝐴 ) = 𝐵 → ( ( 𝐹 ‘ 𝐴 ) ≠ ∅ → 𝐴 𝐹 ( 𝐹 ‘ 𝐴 ) ) )
15		neeq1	⊢ ( ( 𝐹 ‘ 𝐴 ) = 𝐵 → ( ( 𝐹 ‘ 𝐴 ) ≠ ∅ ↔ 𝐵 ≠ ∅ ) )
16		breq2	⊢ ( ( 𝐹 ‘ 𝐴 ) = 𝐵 → ( 𝐴 𝐹 ( 𝐹 ‘ 𝐴 ) ↔ 𝐴 𝐹 𝐵 ) )
17	14 15 16	3imtr3d	⊢ ( ( 𝐹 ‘ 𝐴 ) = 𝐵 → ( 𝐵 ≠ ∅ → 𝐴 𝐹 𝐵 ) )
18	17	com12	⊢ ( 𝐵 ≠ ∅ → ( ( 𝐹 ‘ 𝐴 ) = 𝐵 → 𝐴 𝐹 𝐵 ) )