Theorem tz6.12i 6908

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

Assertion

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

Proof of Theorem tz6.12i

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6895	. . . . 5 ⊢ (𝐹‘𝐴) ∈ V
2		neeq1 3026	. . . . . . . 8 ⊢ ((𝐹‘𝐴) = 𝑦 → ((𝐹‘𝐴) ≠ ∅ ↔ 𝑦 ≠ ∅))
3		tz6.12-2 6869	. . . . . . . . . . 11 ⊢ (¬ ∃!𝑦 𝐴𝐹𝑦 → (𝐹‘𝐴) = ∅)
4	3	necon1ai 2991	. . . . . . . . . 10 ⊢ ((𝐹‘𝐴) ≠ ∅ → ∃!𝑦 𝐴𝐹𝑦)
5		tz6.12c 6904	. . . . . . . . . 10 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))
6	4, 5	syl 18	. . . . . . . . 9 ⊢ ((𝐹‘𝐴) ≠ ∅ → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))
7	6	biimpcd 252	. . . . . . . 8 ⊢ ((𝐹‘𝐴) = 𝑦 → ((𝐹‘𝐴) ≠ ∅ → 𝐴𝐹𝑦))
8	2, 7	sylbird 263	. . . . . . 7 ⊢ ((𝐹‘𝐴) = 𝑦 → (𝑦 ≠ ∅ → 𝐴𝐹𝑦))
9	8	eqcoms 2777	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝐴) → (𝑦 ≠ ∅ → 𝐴𝐹𝑦))
10		neeq1 3026	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝐴) → (𝑦 ≠ ∅ ↔ (𝐹‘𝐴) ≠ ∅))
11		breq2 5115	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝐴) → (𝐴𝐹𝑦 ↔ 𝐴𝐹(𝐹‘𝐴)))
12	9, 10, 11	3imtr3d 296	. . . . 5 ⊢ (𝑦 = (𝐹‘𝐴) → ((𝐹‘𝐴) ≠ ∅ → 𝐴𝐹(𝐹‘𝐴)))
13	1, 12	vtocle 3530	. . . 4 ⊢ ((𝐹‘𝐴) ≠ ∅ → 𝐴𝐹(𝐹‘𝐴))
14	13	a1i 11	. . 3 ⊢ ((𝐹‘𝐴) = 𝐵 → ((𝐹‘𝐴) ≠ ∅ → 𝐴𝐹(𝐹‘𝐴)))
15		neeq1 3026	. . 3 ⊢ ((𝐹‘𝐴) = 𝐵 → ((𝐹‘𝐴) ≠ ∅ ↔ 𝐵 ≠ ∅))
16		breq2 5115	. . 3 ⊢ ((𝐹‘𝐴) = 𝐵 → (𝐴𝐹(𝐹‘𝐴) ↔ 𝐴𝐹𝐵))
17	14, 15, 16	3imtr3d 296	. 2 ⊢ ((𝐹‘𝐴) = 𝐵 → (𝐵 ≠ ∅ → 𝐴𝐹𝐵))
18	17	com12 33	1 ⊢ (𝐵 ≠ ∅ → ((𝐹‘𝐴) = 𝐵 → 𝐴𝐹𝐵))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1567  ∃!weu 2602   ≠ wne 2964  ∅c0 4292   class class class wbr 5111  ‘cfv 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741  ax-nul 5271

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5112  df-iota 6493  df-fv 6545

This theorem is referenced by:  fvbr0  6909  fvclss  7240  dcomex  10431