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Theorem tz6.12i 6930
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 6915 . . . . 5 (𝐹𝐴) ∈ V
2 neeq1 3000 . . . . . . . 8 ((𝐹𝐴) = 𝑦 → ((𝐹𝐴) ≠ ∅ ↔ 𝑦 ≠ ∅))
3 tz6.12-2 6890 . . . . . . . . . . 11 (¬ ∃!𝑦 𝐴𝐹𝑦 → (𝐹𝐴) = ∅)
43necon1ai 2965 . . . . . . . . . 10 ((𝐹𝐴) ≠ ∅ → ∃!𝑦 𝐴𝐹𝑦)
5 tz6.12c 6924 . . . . . . . . . 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 2736 . . . . . 6 (𝑦 = (𝐹𝐴) → (𝑦 ≠ ∅ → 𝐴𝐹𝑦))
10 neeq1 3000 . . . . . 6 (𝑦 = (𝐹𝐴) → (𝑦 ≠ ∅ ↔ (𝐹𝐴) ≠ ∅))
11 breq2 5156 . . . . . 6 (𝑦 = (𝐹𝐴) → (𝐴𝐹𝑦𝐴𝐹(𝐹𝐴)))
129, 10, 113imtr3d 292 . . . . 5 (𝑦 = (𝐹𝐴) → ((𝐹𝐴) ≠ ∅ → 𝐴𝐹(𝐹𝐴)))
131, 12vtocle 3543 . . . 4 ((𝐹𝐴) ≠ ∅ → 𝐴𝐹(𝐹𝐴))
1413a1i 11 . . 3 ((𝐹𝐴) = 𝐵 → ((𝐹𝐴) ≠ ∅ → 𝐴𝐹(𝐹𝐴)))
15 neeq1 3000 . . 3 ((𝐹𝐴) = 𝐵 → ((𝐹𝐴) ≠ ∅ ↔ 𝐵 ≠ ∅))
16 breq2 5156 . . 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 1533  ∃!weu 2557  wne 2937  c0 4326   class class class wbr 5152  cfv 6553
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 2166  ax-ext 2699  ax-nul 5310
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561
This theorem is referenced by:  fvbr0  6931  fvclss  7257  dcomex  10478
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