Theorem tz6.12i 6695

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 6682	. . . . 5 ⊢ (𝐹‘𝐴) ∈ V
2		neeq1 3078	. . . . . . . 8 ⊢ ((𝐹‘𝐴) = 𝑦 → ((𝐹‘𝐴) ≠ ∅ ↔ 𝑦 ≠ ∅))
3		tz6.12-2 6659	. . . . . . . . . . 11 ⊢ (¬ ∃!𝑦 𝐴𝐹𝑦 → (𝐹‘𝐴) = ∅)
4	3	necon1ai 3043	. . . . . . . . . 10 ⊢ ((𝐹‘𝐴) ≠ ∅ → ∃!𝑦 𝐴𝐹𝑦)
5		tz6.12c 6694	. . . . . . . . . 10 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))
6	4, 5	syl 17	. . . . . . . . 9 ⊢ ((𝐹‘𝐴) ≠ ∅ → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))
7	6	biimpcd 251	. . . . . . . 8 ⊢ ((𝐹‘𝐴) = 𝑦 → ((𝐹‘𝐴) ≠ ∅ → 𝐴𝐹𝑦))
8	2, 7	sylbird 262	. . . . . . 7 ⊢ ((𝐹‘𝐴) = 𝑦 → (𝑦 ≠ ∅ → 𝐴𝐹𝑦))
9	8	eqcoms 2829	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝐴) → (𝑦 ≠ ∅ → 𝐴𝐹𝑦))
10		neeq1 3078	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝐴) → (𝑦 ≠ ∅ ↔ (𝐹‘𝐴) ≠ ∅))
11		breq2 5069	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝐴) → (𝐴𝐹𝑦 ↔ 𝐴𝐹(𝐹‘𝐴)))
12	9, 10, 11	3imtr3d 295	. . . . 5 ⊢ (𝑦 = (𝐹‘𝐴) → ((𝐹‘𝐴) ≠ ∅ → 𝐴𝐹(𝐹‘𝐴)))
13	1, 12	vtocle 3583	. . . 4 ⊢ ((𝐹‘𝐴) ≠ ∅ → 𝐴𝐹(𝐹‘𝐴))
14	13	a1i 11	. . 3 ⊢ ((𝐹‘𝐴) = 𝐵 → ((𝐹‘𝐴) ≠ ∅ → 𝐴𝐹(𝐹‘𝐴)))
15		neeq1 3078	. . 3 ⊢ ((𝐹‘𝐴) = 𝐵 → ((𝐹‘𝐴) ≠ ∅ ↔ 𝐵 ≠ ∅))
16		breq2 5069	. . 3 ⊢ ((𝐹‘𝐴) = 𝐵 → (𝐴𝐹(𝐹‘𝐴) ↔ 𝐴𝐹𝐵))
17	14, 15, 16	3imtr3d 295	. 2 ⊢ ((𝐹‘𝐴) = 𝐵 → (𝐵 ≠ ∅ → 𝐴𝐹𝐵))
18	17	com12 32	1 ⊢ (𝐵 ≠ ∅ → ((𝐹‘𝐴) = 𝐵 → 𝐴𝐹𝐵))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1533  ∃!weu 2649   ≠ wne 3016  ∅c0 4290   class class class wbr 5065  ‘cfv 6354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-nul 5209

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-iota 6313  df-fv 6362

This theorem is referenced by:  fvbr0  6696  fvclss  7000  dcomex  9868