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

Theorem brcogw 5323
Description: Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)
Assertion
Ref Expression
brcogw (((𝐴𝑉𝐵𝑊𝑋𝑍) ∧ (𝐴𝐷𝑋𝑋𝐶𝐵)) → 𝐴(𝐶𝐷)𝐵)

Proof of Theorem brcogw
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 3simpa 1078 . 2 ((𝐴𝑉𝐵𝑊𝑋𝑍) → (𝐴𝑉𝐵𝑊))
2 breq2 4689 . . . . . 6 (𝑥 = 𝑋 → (𝐴𝐷𝑥𝐴𝐷𝑋))
3 breq1 4688 . . . . . 6 (𝑥 = 𝑋 → (𝑥𝐶𝐵𝑋𝐶𝐵))
42, 3anbi12d 747 . . . . 5 (𝑥 = 𝑋 → ((𝐴𝐷𝑥𝑥𝐶𝐵) ↔ (𝐴𝐷𝑋𝑋𝐶𝐵)))
54spcegv 3325 . . . 4 (𝑋𝑍 → ((𝐴𝐷𝑋𝑋𝐶𝐵) → ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
65imp 444 . . 3 ((𝑋𝑍 ∧ (𝐴𝐷𝑋𝑋𝐶𝐵)) → ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵))
763ad2antl3 1245 . 2 (((𝐴𝑉𝐵𝑊𝑋𝑍) ∧ (𝐴𝐷𝑋𝑋𝐶𝐵)) → ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵))
8 brcog 5321 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
98biimpar 501 . 2 (((𝐴𝑉𝐵𝑊) ∧ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)) → 𝐴(𝐶𝐷)𝐵)
101, 7, 9syl2an2r 893 1 (((𝐴𝑉𝐵𝑊𝑋𝑍) ∧ (𝐴𝐷𝑋𝑋𝐶𝐵)) → 𝐴(𝐶𝐷)𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030   class class class wbr 4685  ccom 5147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-co 5152
This theorem is referenced by:  utop2nei  22101  utop3cls  22102  iunrelexpuztr  38328  frege96d  38358  frege98d  38362
  Copyright terms: Public domain W3C validator