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Theorem eusv2nf 4834
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 2479 . . . 4 𝑦∃!𝑦𝑥 𝑦 = 𝐴
2 nfe1 2024 . . . . . . 7 𝑥𝑥 𝑦 = 𝐴
32nfeu 2485 . . . . . 6 𝑥∃!𝑦𝑥 𝑦 = 𝐴
4 eusv2.1 . . . . . . . . 9 𝐴 ∈ V
54isseti 3199 . . . . . . . 8 𝑦 𝑦 = 𝐴
6 19.8a 2049 . . . . . . . . 9 (𝑦 = 𝐴 → ∃𝑥 𝑦 = 𝐴)
76ancri 574 . . . . . . . 8 (𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴𝑦 = 𝐴))
85, 7eximii 1761 . . . . . . 7 𝑦(∃𝑥 𝑦 = 𝐴𝑦 = 𝐴)
9 eupick 2535 . . . . . . 7 ((∃!𝑦𝑥 𝑦 = 𝐴 ∧ ∃𝑦(∃𝑥 𝑦 = 𝐴𝑦 = 𝐴)) → (∃𝑥 𝑦 = 𝐴𝑦 = 𝐴))
108, 9mpan2 706 . . . . . 6 (∃!𝑦𝑥 𝑦 = 𝐴 → (∃𝑥 𝑦 = 𝐴𝑦 = 𝐴))
113, 10alrimi 2080 . . . . 5 (∃!𝑦𝑥 𝑦 = 𝐴 → ∀𝑥(∃𝑥 𝑦 = 𝐴𝑦 = 𝐴))
12 nf6 2114 . . . . 5 (Ⅎ𝑥 𝑦 = 𝐴 ↔ ∀𝑥(∃𝑥 𝑦 = 𝐴𝑦 = 𝐴))
1311, 12sylibr 224 . . . 4 (∃!𝑦𝑥 𝑦 = 𝐴 → Ⅎ𝑥 𝑦 = 𝐴)
141, 13alrimi 2080 . . 3 (∃!𝑦𝑥 𝑦 = 𝐴 → ∀𝑦𝑥 𝑦 = 𝐴)
15 dfnfc2 4427 . . . 4 (∀𝑥 𝐴 ∈ V → (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦 = 𝐴))
1615, 4mpg 1721 . . 3 (𝑥𝐴 ↔ ∀𝑦𝑥 𝑦 = 𝐴)
1714, 16sylibr 224 . 2 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
18 eusvnfb 4832 . . . 4 (∃!𝑦𝑥 𝑦 = 𝐴 ↔ (𝑥𝐴𝐴 ∈ V))
194, 18mpbiran2 953 . . 3 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
20 eusv2i 4833 . . 3 (∃!𝑦𝑥 𝑦 = 𝐴 → ∃!𝑦𝑥 𝑦 = 𝐴)
2119, 20sylbir 225 . 2 (𝑥𝐴 → ∃!𝑦𝑥 𝑦 = 𝐴)
2217, 21impbii 199 1 (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wex 1701  wnf 1705  wcel 1987  ∃!weu 2469  wnfc 2748  Vcvv 3190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-nul 3898  df-sn 4156  df-pr 4158  df-uni 4410
This theorem is referenced by:  eusv2  4835
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