Theorem tz6.12i 6912

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 6897	. . . . 5 ⊢ (𝐹‘𝐴) ∈ V
2		neeq1 2997	. . . . . . . 8 ⊢ ((𝐹‘𝐴) = 𝑦 → ((𝐹‘𝐴) ≠ ∅ ↔ 𝑦 ≠ ∅))
3		tz6.12-2 6872	. . . . . . . . . . 11 ⊢ (¬ ∃!𝑦 𝐴𝐹𝑦 → (𝐹‘𝐴) = ∅)
4	3	necon1ai 2962	. . . . . . . . . 10 ⊢ ((𝐹‘𝐴) ≠ ∅ → ∃!𝑦 𝐴𝐹𝑦)
5		tz6.12c 6906	. . . . . . . . . 10 ⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))
6	4, 5	syl 17	. . . . . . . . 9 ⊢ ((𝐹‘𝐴) ≠ ∅ → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))
7	6	biimpcd 248	. . . . . . . 8 ⊢ ((𝐹‘𝐴) = 𝑦 → ((𝐹‘𝐴) ≠ ∅ → 𝐴𝐹𝑦))
8	2, 7	sylbird 260	. . . . . . 7 ⊢ ((𝐹‘𝐴) = 𝑦 → (𝑦 ≠ ∅ → 𝐴𝐹𝑦))
9	8	eqcoms 2734	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝐴) → (𝑦 ≠ ∅ → 𝐴𝐹𝑦))
10		neeq1 2997	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝐴) → (𝑦 ≠ ∅ ↔ (𝐹‘𝐴) ≠ ∅))
11		breq2 5145	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝐴) → (𝐴𝐹𝑦 ↔ 𝐴𝐹(𝐹‘𝐴)))
12	9, 10, 11	3imtr3d 293	. . . . 5 ⊢ (𝑦 = (𝐹‘𝐴) → ((𝐹‘𝐴) ≠ ∅ → 𝐴𝐹(𝐹‘𝐴)))
13	1, 12	vtocle 3538	. . . 4 ⊢ ((𝐹‘𝐴) ≠ ∅ → 𝐴𝐹(𝐹‘𝐴))
14	13	a1i 11	. . 3 ⊢ ((𝐹‘𝐴) = 𝐵 → ((𝐹‘𝐴) ≠ ∅ → 𝐴𝐹(𝐹‘𝐴)))
15		neeq1 2997	. . 3 ⊢ ((𝐹‘𝐴) = 𝐵 → ((𝐹‘𝐴) ≠ ∅ ↔ 𝐵 ≠ ∅))
16		breq2 5145	. . 3 ⊢ ((𝐹‘𝐴) = 𝐵 → (𝐴𝐹(𝐹‘𝐴) ↔ 𝐴𝐹𝐵))
17	14, 15, 16	3imtr3d 293	. 2 ⊢ ((𝐹‘𝐴) = 𝐵 → (𝐵 ≠ ∅ → 𝐴𝐹𝐵))
18	17	com12 32	1 ⊢ (𝐵 ≠ ∅ → ((𝐹‘𝐴) = 𝐵 → 𝐴𝐹𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533  ∃!weu 2556   ≠ wne 2934  ∅c0 4317   class class class wbr 5141  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-nul 5299

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544

This theorem is referenced by:  fvbr0  6913  fvclss  7235  dcomex  10441