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Theorem eusv2nf 4416
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 2017 . . . 4 𝑦∃!𝑦𝑥 𝑦 = 𝐴
2 nfe1 1476 . . . . . . 7 𝑥𝑥 𝑦 = 𝐴
32nfeu 2025 . . . . . 6 𝑥∃!𝑦𝑥 𝑦 = 𝐴
4 eusv2.1 . . . . . . . . 9 𝐴 ∈ V
54isseti 2720 . . . . . . . 8 𝑦 𝑦 = 𝐴
6 19.8a 1570 . . . . . . . . 9 (𝑦 = 𝐴 → ∃𝑥 𝑦 = 𝐴)
76ancri 322 . . . . . . . 8 (𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴𝑦 = 𝐴))
85, 7eximii 1582 . . . . . . 7 𝑦(∃𝑥 𝑦 = 𝐴𝑦 = 𝐴)
9 eupick 2085 . . . . . . 7 ((∃!𝑦𝑥 𝑦 = 𝐴 ∧ ∃𝑦(∃𝑥 𝑦 = 𝐴𝑦 = 𝐴)) → (∃𝑥 𝑦 = 𝐴𝑦 = 𝐴))
108, 9mpan2 422 . . . . . 6 (∃!𝑦𝑥 𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴𝑦 = 𝐴))
113, 10alrimi 1502 . . . . 5 (∃!𝑦𝑥 𝑦 = 𝐴 → ∀𝑥(∃𝑥 𝑦 = 𝐴𝑦 = 𝐴))
12 nf3 1649 . . . . 5 (Ⅎ𝑥 𝑦 = 𝐴 ↔ ∀𝑥(∃𝑥 𝑦 = 𝐴𝑦 = 𝐴))
1311, 12sylibr 133 . . . 4 (∃!𝑦𝑥 𝑦 = 𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
141, 13alrimi 1502 . . 3 (∃!𝑦𝑥 𝑦 = 𝐴 → ∀𝑦𝑥 𝑦 = 𝐴)
15 dfnfc2 3790 . . . 4 (∀𝑥 𝐴 ∈ V → (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦 = 𝐴))
1615, 4mpg 1431 . . 3 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦 = 𝐴)
1714, 16sylibr 133 . 2 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
18 eusvnfb 4414 . . . 4 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ (𝑥𝐴𝐴 ∈ V))
194, 18mpbiran2 926 . . 3 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
20 eusv2i 4415 . . 3 (∃!𝑦𝑥 𝑦 = 𝐴 → ∃!𝑦𝑥 𝑦 = 𝐴)
2119, 20sylbir 134 . 2 (𝑥𝐴 → ∃!𝑦𝑥 𝑦 = 𝐴)
2217, 21impbii 125 1 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1333   = wceq 1335  wnf 1440  wex 1472  ∃!weu 2006  wcel 2128  wnfc 2286  Vcvv 2712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-sn 3566  df-pr 3567  df-uni 3773
This theorem is referenced by:  eusv2  4417
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