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Theorem tz6.12i 6689
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 6676 . . . . 5 (𝐹𝐴) ∈ V
2 neeq1 3075 . . . . . . . 8 ((𝐹𝐴) = 𝑦 → ((𝐹𝐴) ≠ ∅ ↔ 𝑦 ≠ ∅))
3 tz6.12-2 6653 . . . . . . . . . . 11 (¬ ∃!𝑦 𝐴𝐹𝑦 → (𝐹𝐴) = ∅)
43necon1ai 3040 . . . . . . . . . 10 ((𝐹𝐴) ≠ ∅ → ∃!𝑦 𝐴𝐹𝑦)
5 tz6.12c 6688 . . . . . . . . . 10 (∃!𝑦 𝐴𝐹𝑦 → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))
64, 5syl 17 . . . . . . . . 9 ((𝐹𝐴) ≠ ∅ → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))
76biimpcd 250 . . . . . . . 8 ((𝐹𝐴) = 𝑦 → ((𝐹𝐴) ≠ ∅ → 𝐴𝐹𝑦))
82, 7sylbird 261 . . . . . . 7 ((𝐹𝐴) = 𝑦 → (𝑦 ≠ ∅ → 𝐴𝐹𝑦))
98eqcoms 2826 . . . . . 6 (𝑦 = (𝐹𝐴) → (𝑦 ≠ ∅ → 𝐴𝐹𝑦))
10 neeq1 3075 . . . . . 6 (𝑦 = (𝐹𝐴) → (𝑦 ≠ ∅ ↔ (𝐹𝐴) ≠ ∅))
11 breq2 5061 . . . . . 6 (𝑦 = (𝐹𝐴) → (𝐴𝐹𝑦𝐴𝐹(𝐹𝐴)))
129, 10, 113imtr3d 294 . . . . 5 (𝑦 = (𝐹𝐴) → ((𝐹𝐴) ≠ ∅ → 𝐴𝐹(𝐹𝐴)))
131, 12vtocle 3581 . . . 4 ((𝐹𝐴) ≠ ∅ → 𝐴𝐹(𝐹𝐴))
1413a1i 11 . . 3 ((𝐹𝐴) = 𝐵 → ((𝐹𝐴) ≠ ∅ → 𝐴𝐹(𝐹𝐴)))
15 neeq1 3075 . . 3 ((𝐹𝐴) = 𝐵 → ((𝐹𝐴) ≠ ∅ ↔ 𝐵 ≠ ∅))
16 breq2 5061 . . 3 ((𝐹𝐴) = 𝐵 → (𝐴𝐹(𝐹𝐴) ↔ 𝐴𝐹𝐵))
1714, 15, 163imtr3d 294 . 2 ((𝐹𝐴) = 𝐵 → (𝐵 ≠ ∅ → 𝐴𝐹𝐵))
1817com12 32 1 (𝐵 ≠ ∅ → ((𝐹𝐴) = 𝐵𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  ∃!weu 2646  wne 3013  c0 4288   class class class wbr 5057  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356
This theorem is referenced by:  fvbr0  6690  fvclss  6992  dcomex  9857
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