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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 3076 . . . . . . . 8 ((𝐹𝐴) = 𝑦 → ((𝐹𝐴) ≠ ∅ ↔ 𝑦 ≠ ∅))
3 tz6.12-2 6653 . . . . . . . . . . 11 (¬ ∃!𝑦 𝐴𝐹𝑦 → (𝐹𝐴) = ∅)
43necon1ai 3041 . . . . . . . . . 10 ((𝐹𝐴) ≠ ∅ → ∃!𝑦 𝐴𝐹𝑦)
5 tz6.12c 6688 . . . . . . . . . 10 (∃!𝑦 𝐴𝐹𝑦 → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))
64, 5syl 17 . . . . . . . . 9 ((𝐹𝐴) ≠ ∅ → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))
76biimpcd 251 . . . . . . . 8 ((𝐹𝐴) = 𝑦 → ((𝐹𝐴) ≠ ∅ → 𝐴𝐹𝑦))
82, 7sylbird 262 . . . . . . 7 ((𝐹𝐴) = 𝑦 → (𝑦 ≠ ∅ → 𝐴𝐹𝑦))
98eqcoms 2827 . . . . . 6 (𝑦 = (𝐹𝐴) → (𝑦 ≠ ∅ → 𝐴𝐹𝑦))
10 neeq1 3076 . . . . . 6 (𝑦 = (𝐹𝐴) → (𝑦 ≠ ∅ ↔ (𝐹𝐴) ≠ ∅))
11 breq2 5061 . . . . . 6 (𝑦 = (𝐹𝐴) → (𝐴𝐹𝑦𝐴𝐹(𝐹𝐴)))
129, 10, 113imtr3d 295 . . . . 5 (𝑦 = (𝐹𝐴) → ((𝐹𝐴) ≠ ∅ → 𝐴𝐹(𝐹𝐴)))
131, 12vtocle 3582 . . . 4 ((𝐹𝐴) ≠ ∅ → 𝐴𝐹(𝐹𝐴))
1413a1i 11 . . 3 ((𝐹𝐴) = 𝐵 → ((𝐹𝐴) ≠ ∅ → 𝐴𝐹(𝐹𝐴)))
15 neeq1 3076 . . 3 ((𝐹𝐴) = 𝐵 → ((𝐹𝐴) ≠ ∅ ↔ 𝐵 ≠ ∅))
16 breq2 5061 . . 3 ((𝐹𝐴) = 𝐵 → (𝐴𝐹(𝐹𝐴) ↔ 𝐴𝐹𝐵))
1714, 15, 163imtr3d 295 . 2 ((𝐹𝐴) = 𝐵 → (𝐵 ≠ ∅ → 𝐴𝐹𝐵))
1817com12 32 1 (𝐵 ≠ ∅ → ((𝐹𝐴) = 𝐵𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208   = wceq 1530  ∃!weu 2647  wne 3014  c0 4289   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 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-nul 5201
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356
This theorem is referenced by:  fvbr0  6690  fvclss  6993  dcomex  9861
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