MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfid3 Structured version   Visualization version   GIF version

Theorem dfid3 5310
Description: A stronger version of df-id 5309 that does not require 𝑥 and 𝑦 to be disjoint. This is not the "official" definition since our definition soundness check without this requirement would be much more complex. The proof can be instructive in showing how disjoint variable requirements may be eliminated, a task that is not necessarily obvious. (Contributed by NM, 5-Feb-2008.) (Revised by Mario Carneiro, 18-Nov-2016.) Use directly the definition df-id 5309 when sufficient, since the derivation of dfid3 5310 is nontrivial and uses auxiliary axioms ax-10 2080 to ax-13 2302. (New usage is discouraged.)
Assertion
Ref Expression
dfid3 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}

Proof of Theorem dfid3
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-id 5309 . 2 I = {⟨𝑥, 𝑧⟩ ∣ 𝑥 = 𝑧}
2 equcom 1976 . . . . . . . . . . 11 (𝑥 = 𝑧𝑧 = 𝑥)
32anbi1ci 617 . . . . . . . . . 10 ((𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ (𝑧 = 𝑥𝑤 = ⟨𝑥, 𝑧⟩))
43exbii 1811 . . . . . . . . 9 (∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑧(𝑧 = 𝑥𝑤 = ⟨𝑥, 𝑧⟩))
5 opeq2 4675 . . . . . . . . . . 11 (𝑧 = 𝑥 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝑥⟩)
65eqeq2d 2783 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑤 = ⟨𝑥, 𝑧⟩ ↔ 𝑤 = ⟨𝑥, 𝑥⟩))
76equsexvw 1963 . . . . . . . . 9 (∃𝑧(𝑧 = 𝑥𝑤 = ⟨𝑥, 𝑧⟩) ↔ 𝑤 = ⟨𝑥, 𝑥⟩)
8 equid 1970 . . . . . . . . . 10 𝑥 = 𝑥
98biantru 522 . . . . . . . . 9 (𝑤 = ⟨𝑥, 𝑥⟩ ↔ (𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
104, 7, 93bitri 289 . . . . . . . 8 (∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ (𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
1110exbii 1811 . . . . . . 7 (∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
12 nfe1 2088 . . . . . . . 8 𝑥𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥)
131219.9 2135 . . . . . . 7 (∃𝑥𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ ∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
1411, 13bitr4i 270 . . . . . 6 (∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑥𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥))
15 opeq2 4675 . . . . . . . . . . 11 (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑥, 𝑦⟩)
1615eqeq2d 2783 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑤 = ⟨𝑥, 𝑥⟩ ↔ 𝑤 = ⟨𝑥, 𝑦⟩))
17 equequ2 1984 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥 = 𝑥𝑥 = 𝑦))
1816, 17anbi12d 622 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
1918sps 2114 . . . . . . . 8 (∀𝑥 𝑥 = 𝑦 → ((𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
2019drex1 2378 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
2120drex2 2379 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑥(𝑤 = ⟨𝑥, 𝑥⟩ ∧ 𝑥 = 𝑥) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
2214, 21syl5bb 275 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
23 nfnae 2371 . . . . . 6 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
24 nfnae 2371 . . . . . . 7 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
25 nfcvd 2928 . . . . . . . . 9 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑤)
26 nfcvf2 2954 . . . . . . . . . 10 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)
27 nfcvd 2928 . . . . . . . . . 10 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑧)
2826, 27nfopd 4691 . . . . . . . . 9 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥, 𝑧⟩)
2925, 28nfeqd 2935 . . . . . . . 8 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑤 = ⟨𝑥, 𝑧⟩)
3026, 27nfeqd 2935 . . . . . . . 8 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦 𝑥 = 𝑧)
3129, 30nfand 1861 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧))
32 opeq2 4675 . . . . . . . . . 10 (𝑧 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑥, 𝑦⟩)
3332eqeq2d 2783 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑤 = ⟨𝑥, 𝑧⟩ ↔ 𝑤 = ⟨𝑥, 𝑦⟩))
34 equequ2 1984 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑥 = 𝑧𝑥 = 𝑦))
3533, 34anbi12d 622 . . . . . . . 8 (𝑧 = 𝑦 → ((𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
3635a1i 11 . . . . . . 7 (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ((𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦))))
3724, 31, 36cbvexd 2343 . . . . . 6 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
3823, 37exbid 2156 . . . . 5 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)))
3922, 38pm2.61i 177 . . . 4 (∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧) ↔ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦))
4039abbii 2839 . . 3 {𝑤 ∣ ∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧)} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)}
41 df-opab 4989 . . 3 {⟨𝑥, 𝑧⟩ ∣ 𝑥 = 𝑧} = {𝑤 ∣ ∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝑥 = 𝑧)}
42 df-opab 4989 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝑥 = 𝑦)}
4340, 41, 423eqtr4i 2807 . 2 {⟨𝑥, 𝑧⟩ ∣ 𝑥 = 𝑧} = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
441, 43eqtri 2797 1 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  wal 1506   = wceq 1508  wex 1743  {cab 2753  cop 4442  {copab 4988   I cid 5308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-rab 3092  df-v 3412  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-sn 4437  df-pr 4439  df-op 4443  df-opab 4989  df-id 5309
This theorem is referenced by:  dfid2  5312
  Copyright terms: Public domain W3C validator