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Theorem eusv2nf 4216
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 1927 . . . 4 𝑦∃!𝑦𝑥 𝑦 = 𝐴
2 nfe1 1401 . . . . . . 7 𝑥𝑥 𝑦 = 𝐴
32nfeu 1935 . . . . . 6 𝑥∃!𝑦𝑥 𝑦 = 𝐴
4 eusv2.1 . . . . . . . . 9 𝐴 ∈ V
54isseti 2580 . . . . . . . 8 𝑦 𝑦 = 𝐴
6 19.8a 1498 . . . . . . . . 9 (𝑦 = 𝐴 → ∃𝑥 𝑦 = 𝐴)
76ancri 311 . . . . . . . 8 (𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴𝑦 = 𝐴))
85, 7eximii 1509 . . . . . . 7 𝑦(∃𝑥 𝑦 = 𝐴𝑦 = 𝐴)
9 eupick 1995 . . . . . . 7 ((∃!𝑦𝑥 𝑦 = 𝐴 ∧ ∃𝑦(∃𝑥 𝑦 = 𝐴𝑦 = 𝐴)) → (∃𝑥 𝑦 = 𝐴𝑦 = 𝐴))
108, 9mpan2 409 . . . . . 6 (∃!𝑦𝑥 𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴𝑦 = 𝐴))
113, 10alrimi 1431 . . . . 5 (∃!𝑦𝑥 𝑦 = 𝐴 → ∀𝑥(∃𝑥 𝑦 = 𝐴𝑦 = 𝐴))
12 nf3 1575 . . . . 5 (Ⅎ𝑥 𝑦 = 𝐴 ↔ ∀𝑥(∃𝑥 𝑦 = 𝐴𝑦 = 𝐴))
1311, 12sylibr 141 . . . 4 (∃!𝑦𝑥 𝑦 = 𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
141, 13alrimi 1431 . . 3 (∃!𝑦𝑥 𝑦 = 𝐴 → ∀𝑦𝑥 𝑦 = 𝐴)
15 dfnfc2 3626 . . . 4 (∀𝑥 𝐴 ∈ V → (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦 = 𝐴))
1615, 4mpg 1356 . . 3 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦 = 𝐴)
1714, 16sylibr 141 . 2 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
18 eusvnfb 4214 . . . 4 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ (𝑥𝐴𝐴 ∈ V))
194, 18mpbiran2 859 . . 3 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
20 eusv2i 4215 . . 3 (∃!𝑦𝑥 𝑦 = 𝐴 → ∃!𝑦𝑥 𝑦 = 𝐴)
2119, 20sylbir 129 . 2 (𝑥𝐴 → ∃!𝑦𝑥 𝑦 = 𝐴)
2217, 21impbii 121 1 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257   = wceq 1259  wnf 1365  wex 1397  wcel 1409  ∃!weu 1916  wnfc 2181  Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-sn 3409  df-pr 3410  df-uni 3609
This theorem is referenced by:  eusv2  4217
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