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Theorem tz6.12i 6376
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 6363 . . . . 5 (𝐹𝐴) ∈ V
2 neeq1 2994 . . . . . . . 8 ((𝐹𝐴) = 𝑦 → ((𝐹𝐴) ≠ ∅ ↔ 𝑦 ≠ ∅))
3 tz6.12-2 6344 . . . . . . . . . . 11 (¬ ∃!𝑦 𝐴𝐹𝑦 → (𝐹𝐴) = ∅)
43necon1ai 2959 . . . . . . . . . 10 ((𝐹𝐴) ≠ ∅ → ∃!𝑦 𝐴𝐹𝑦)
5 tz6.12c 6375 . . . . . . . . . 10 (∃!𝑦 𝐴𝐹𝑦 → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))
64, 5syl 17 . . . . . . . . 9 ((𝐹𝐴) ≠ ∅ → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))
76biimpcd 239 . . . . . . . 8 ((𝐹𝐴) = 𝑦 → ((𝐹𝐴) ≠ ∅ → 𝐴𝐹𝑦))
82, 7sylbird 250 . . . . . . 7 ((𝐹𝐴) = 𝑦 → (𝑦 ≠ ∅ → 𝐴𝐹𝑦))
98eqcoms 2768 . . . . . 6 (𝑦 = (𝐹𝐴) → (𝑦 ≠ ∅ → 𝐴𝐹𝑦))
10 neeq1 2994 . . . . . 6 (𝑦 = (𝐹𝐴) → (𝑦 ≠ ∅ ↔ (𝐹𝐴) ≠ ∅))
11 breq2 4808 . . . . . 6 (𝑦 = (𝐹𝐴) → (𝐴𝐹𝑦𝐴𝐹(𝐹𝐴)))
129, 10, 113imtr3d 282 . . . . 5 (𝑦 = (𝐹𝐴) → ((𝐹𝐴) ≠ ∅ → 𝐴𝐹(𝐹𝐴)))
131, 12vtocle 3422 . . . 4 ((𝐹𝐴) ≠ ∅ → 𝐴𝐹(𝐹𝐴))
1413a1i 11 . . 3 ((𝐹𝐴) = 𝐵 → ((𝐹𝐴) ≠ ∅ → 𝐴𝐹(𝐹𝐴)))
15 neeq1 2994 . . 3 ((𝐹𝐴) = 𝐵 → ((𝐹𝐴) ≠ ∅ ↔ 𝐵 ≠ ∅))
16 breq2 4808 . . 3 ((𝐹𝐴) = 𝐵 → (𝐴𝐹(𝐹𝐴) ↔ 𝐴𝐹𝐵))
1714, 15, 163imtr3d 282 . 2 ((𝐹𝐴) = 𝐵 → (𝐵 ≠ ∅ → 𝐴𝐹𝐵))
1817com12 32 1 (𝐵 ≠ ∅ → ((𝐹𝐴) = 𝐵𝐴𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1632  ∃!weu 2607  wne 2932  c0 4058   class class class wbr 4804  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-nul 4941
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057
This theorem is referenced by:  fvbr0  6377  fvclss  6664  dcomex  9481
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