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Theorem eusv2nf 4785
Description: Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by Mario Carneiro, 18-Nov-2016.)
Hypothesis
Ref Expression
eusv2.1 𝐴 ∈ V
Assertion
Ref Expression
eusv2nf (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem eusv2nf
StepHypRef Expression
1 nfeu1 2468 . . . 4 𝑦∃!𝑦𝑥 𝑦 = 𝐴
2 nfe1 2014 . . . . . . 7 𝑥𝑥 𝑦 = 𝐴
32nfeu 2474 . . . . . 6 𝑥∃!𝑦𝑥 𝑦 = 𝐴
4 eusv2.1 . . . . . . . . 9 𝐴 ∈ V
54isseti 3182 . . . . . . . 8 𝑦 𝑦 = 𝐴
6 19.8a 2039 . . . . . . . . 9 (𝑦 = 𝐴 → ∃𝑥 𝑦 = 𝐴)
76ancri 573 . . . . . . . 8 (𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴𝑦 = 𝐴))
85, 7eximii 1754 . . . . . . 7 𝑦(∃𝑥 𝑦 = 𝐴𝑦 = 𝐴)
9 eupick 2524 . . . . . . 7 ((∃!𝑦𝑥 𝑦 = 𝐴 ∧ ∃𝑦(∃𝑥 𝑦 = 𝐴𝑦 = 𝐴)) → (∃𝑥 𝑦 = 𝐴𝑦 = 𝐴))
108, 9mpan2 703 . . . . . 6 (∃!𝑦𝑥 𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴𝑦 = 𝐴))
113, 10alrimi 2069 . . . . 5 (∃!𝑦𝑥 𝑦 = 𝐴 → ∀𝑥(∃𝑥 𝑦 = 𝐴𝑦 = 𝐴))
12 nf6 2103 . . . . 5 (Ⅎ𝑥 𝑦 = 𝐴 ↔ ∀𝑥(∃𝑥 𝑦 = 𝐴𝑦 = 𝐴))
1311, 12sylibr 223 . . . 4 (∃!𝑦𝑥 𝑦 = 𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
141, 13alrimi 2069 . . 3 (∃!𝑦𝑥 𝑦 = 𝐴 → ∀𝑦𝑥 𝑦 = 𝐴)
15 dfnfc2 4385 . . . 4 (∀𝑥 𝐴 ∈ V → (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦 = 𝐴))
1615, 4mpg 1715 . . 3 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦 = 𝐴)
1714, 16sylibr 223 . 2 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
18 eusvnfb 4783 . . . 4 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ (𝑥𝐴𝐴 ∈ V))
194, 18mpbiran2 956 . . 3 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
20 eusv2i 4784 . . 3 (∃!𝑦𝑥 𝑦 = 𝐴 → ∃!𝑦𝑥 𝑦 = 𝐴)
2119, 20sylbir 224 . 2 (𝑥𝐴 → ∃!𝑦𝑥 𝑦 = 𝐴)
2217, 21impbii 198 1 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  wal 1473   = wceq 1475  wex 1695  wnf 1699  wcel 1977  ∃!weu 2458  wnfc 2738  Vcvv 3173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-pr 4128  df-uni 4368
This theorem is referenced by:  eusv2  4786
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