Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-elid3 Structured version   Visualization version   GIF version

Theorem bj-elid3 33398
Description: Characterization of the elements of I. (Contributed by BJ, 29-Mar-2020.)
Assertion
Ref Expression
bj-elid3 (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))

Proof of Theorem bj-elid3
StepHypRef Expression
1 bj-elid 33396 . 2 (⟨𝐴, 𝐵⟩ ∈ I ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)))
2 opelxp 5303 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (V × V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
32anbi1i 733 . . 3 ((⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)))
4 op1stg 7345 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
5 op2ndg 7346 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
64, 5eqeq12d 2775 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩) ↔ 𝐴 = 𝐵))
76pm5.32i 672 . . . 4 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)) ↔ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))
8 simpl 474 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
98anim1i 593 . . . . 5 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐴 = 𝐵))
10 simpl 474 . . . . . 6 ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → 𝐴 ∈ V)
11 eleq1 2827 . . . . . . 7 (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
1211biimpac 504 . . . . . 6 ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → 𝐵 ∈ V)
13 simpr 479 . . . . . 6 ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
1410, 12, 13jca31 558 . . . . 5 ((𝐴 ∈ V ∧ 𝐴 = 𝐵) → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵))
159, 14impbii 199 . . . 4 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ 𝐴 = 𝐵) ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
167, 15bitri 264 . . 3 (((𝐴 ∈ V ∧ 𝐵 ∈ V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)) ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
173, 16bitri 264 . 2 ((⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ (1st ‘⟨𝐴, 𝐵⟩) = (2nd ‘⟨𝐴, 𝐵⟩)) ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
181, 17bitri 264 1 (⟨𝐴, 𝐵⟩ ∈ I ↔ (𝐴 ∈ V ∧ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  cop 4327   I cid 5173   × cxp 5264  cfv 6049  1st c1st 7331  2nd c2nd 7332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fv 6057  df-1st 7333  df-2nd 7334
This theorem is referenced by:  bj-eldiag2  33403
  Copyright terms: Public domain W3C validator