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Theorem tz6.12i 6916
Description: Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
tz6.12i (𝐵 ≠ ∅ → ((𝐹𝐴) = 𝐵𝐴𝐹𝐵))

Proof of Theorem tz6.12i
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fvex 6901 . . . . 5 (𝐹𝐴) ∈ V
2 neeq1 3003 . . . . . . . 8 ((𝐹𝐴) = 𝑦 → ((𝐹𝐴) ≠ ∅ ↔ 𝑦 ≠ ∅))
3 tz6.12-2 6876 . . . . . . . . . . 11 (¬ ∃!𝑦 𝐴𝐹𝑦 → (𝐹𝐴) = ∅)
43necon1ai 2968 . . . . . . . . . 10 ((𝐹𝐴) ≠ ∅ → ∃!𝑦 𝐴𝐹𝑦)
5 tz6.12c 6910 . . . . . . . . . 10 (∃!𝑦 𝐴𝐹𝑦 → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))
64, 5syl 17 . . . . . . . . 9 ((𝐹𝐴) ≠ ∅ → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))
76biimpcd 248 . . . . . . . 8 ((𝐹𝐴) = 𝑦 → ((𝐹𝐴) ≠ ∅ → 𝐴𝐹𝑦))
82, 7sylbird 259 . . . . . . 7 ((𝐹𝐴) = 𝑦 → (𝑦 ≠ ∅ → 𝐴𝐹𝑦))
98eqcoms 2740 . . . . . 6 (𝑦 = (𝐹𝐴) → (𝑦 ≠ ∅ → 𝐴𝐹𝑦))
10 neeq1 3003 . . . . . 6 (𝑦 = (𝐹𝐴) → (𝑦 ≠ ∅ ↔ (𝐹𝐴) ≠ ∅))
11 breq2 5151 . . . . . 6 (𝑦 = (𝐹𝐴) → (𝐴𝐹𝑦𝐴𝐹(𝐹𝐴)))
129, 10, 113imtr3d 292 . . . . 5 (𝑦 = (𝐹𝐴) → ((𝐹𝐴) ≠ ∅ → 𝐴𝐹(𝐹𝐴)))
131, 12vtocle 3575 . . . 4 ((𝐹𝐴) ≠ ∅ → 𝐴𝐹(𝐹𝐴))
1413a1i 11 . . 3 ((𝐹𝐴) = 𝐵 → ((𝐹𝐴) ≠ ∅ → 𝐴𝐹(𝐹𝐴)))
15 neeq1 3003 . . 3 ((𝐹𝐴) = 𝐵 → ((𝐹𝐴) ≠ ∅ ↔ 𝐵 ≠ ∅))
16 breq2 5151 . . 3 ((𝐹𝐴) = 𝐵 → (𝐴𝐹(𝐹𝐴) ↔ 𝐴𝐹𝐵))
1714, 15, 163imtr3d 292 . 2 ((𝐹𝐴) = 𝐵 → (𝐵 ≠ ∅ → 𝐴𝐹𝐵))
1817com12 32 1 (𝐵 ≠ ∅ → ((𝐹𝐴) = 𝐵𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  ∃!weu 2562  wne 2940  c0 4321   class class class wbr 5147  cfv 6540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548
This theorem is referenced by:  fvbr0  6917  fvclss  7237  dcomex  10438
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