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Theorem eusvobj2 7138
Description: Specify the same property in two ways when class 𝐵(𝑦) is single-valued. (Contributed by NM, 1-Nov-2010.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
eusvobj1.1 𝐵 ∈ V
Assertion
Ref Expression
eusvobj2 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (∃𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑥 = 𝐵))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem eusvobj2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 euabsn2 4653 . . 3 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃𝑧{𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧})
2 eleq2 2898 . . . . . 6 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (𝑥 ∈ {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ↔ 𝑥 ∈ {𝑧}))
3 abid 2800 . . . . . 6 (𝑥 ∈ {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ↔ ∃𝑦𝐴 𝑥 = 𝐵)
4 velsn 4573 . . . . . 6 (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
52, 3, 43bitr3g 314 . . . . 5 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦𝐴 𝑥 = 𝐵𝑥 = 𝑧))
6 nfre1 3303 . . . . . . . . 9 𝑦𝑦𝐴 𝑥 = 𝐵
76nfab 2981 . . . . . . . 8 𝑦{𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵}
87nfeq1 2990 . . . . . . 7 𝑦{𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧}
9 eusvobj1.1 . . . . . . . . 9 𝐵 ∈ V
109elabrex 6993 . . . . . . . 8 (𝑦𝐴𝐵 ∈ {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵})
11 eleq2 2898 . . . . . . . . 9 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (𝐵 ∈ {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ↔ 𝐵 ∈ {𝑧}))
129elsn 4572 . . . . . . . . . 10 (𝐵 ∈ {𝑧} ↔ 𝐵 = 𝑧)
13 eqcom 2825 . . . . . . . . . 10 (𝐵 = 𝑧𝑧 = 𝐵)
1412, 13bitri 276 . . . . . . . . 9 (𝐵 ∈ {𝑧} ↔ 𝑧 = 𝐵)
1511, 14syl6bb 288 . . . . . . . 8 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (𝐵 ∈ {𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} ↔ 𝑧 = 𝐵))
1610, 15syl5ib 245 . . . . . . 7 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (𝑦𝐴𝑧 = 𝐵))
178, 16ralrimi 3213 . . . . . 6 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → ∀𝑦𝐴 𝑧 = 𝐵)
18 eqeq1 2822 . . . . . . 7 (𝑥 = 𝑧 → (𝑥 = 𝐵𝑧 = 𝐵))
1918ralbidv 3194 . . . . . 6 (𝑥 = 𝑧 → (∀𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑧 = 𝐵))
2017, 19syl5ibrcom 248 . . . . 5 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (𝑥 = 𝑧 → ∀𝑦𝐴 𝑥 = 𝐵))
215, 20sylbid 241 . . . 4 ({𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
2221exlimiv 1922 . . 3 (∃𝑧{𝑥 ∣ ∃𝑦𝐴 𝑥 = 𝐵} = {𝑧} → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
231, 22sylbi 218 . 2 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (∃𝑦𝐴 𝑥 = 𝐵 → ∀𝑦𝐴 𝑥 = 𝐵))
24 euex 2655 . . 3 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃𝑥𝑦𝐴 𝑥 = 𝐵)
25 rexn0 4450 . . . 4 (∃𝑦𝐴 𝑥 = 𝐵𝐴 ≠ ∅)
2625exlimiv 1922 . . 3 (∃𝑥𝑦𝐴 𝑥 = 𝐵𝐴 ≠ ∅)
27 r19.2z 4436 . . . 4 ((𝐴 ≠ ∅ ∧ ∀𝑦𝐴 𝑥 = 𝐵) → ∃𝑦𝐴 𝑥 = 𝐵)
2827ex 413 . . 3 (𝐴 ≠ ∅ → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
2924, 26, 283syl 18 . 2 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (∀𝑦𝐴 𝑥 = 𝐵 → ∃𝑦𝐴 𝑥 = 𝐵))
3023, 29impbid 213 1 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 → (∃𝑦𝐴 𝑥 = 𝐵 ↔ ∀𝑦𝐴 𝑥 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wex 1771  wcel 2105  ∃!weu 2646  {cab 2796  wne 3013  wral 3135  wrex 3136  Vcvv 3492  c0 4288  {csn 4557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-nul 4289  df-sn 4558
This theorem is referenced by:  eusvobj1  7139
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