Theorem cnvco 4894

Description: Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 19-Mar-1998.) (Revised by set.mm contributors, 27-Aug-2011.)

Assertion

Ref	Expression
cnvco	⊢ ◡(A ∘ B) = (◡B ∘ ◡A)

Proof of Theorem cnvco

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brco 4883	. . . 4 ⊢ (x(A ∘ B)y ↔ ∃z(xBz ∧ zAy))
2		brcnv 4892	. . . . . . 7 ⊢ (z◡Bx ↔ xBz)
3		brcnv 4892	. . . . . . 7 ⊢ (y◡Az ↔ zAy)
4	2, 3	anbi12i 678	. . . . . 6 ⊢ ((z◡Bx ∧ y◡Az) ↔ (xBz ∧ zAy))
5		ancom 437	. . . . . 6 ⊢ ((z◡Bx ∧ y◡Az) ↔ (y◡Az ∧ z◡Bx))
6	4, 5	bitr3i 242	. . . . 5 ⊢ ((xBz ∧ zAy) ↔ (y◡Az ∧ z◡Bx))
7	6	exbii 1582	. . . 4 ⊢ (∃z(xBz ∧ zAy) ↔ ∃z(y◡Az ∧ z◡Bx))
8	1, 7	bitri 240	. . 3 ⊢ (x(A ∘ B)y ↔ ∃z(y◡Az ∧ z◡Bx))
9	8	opabbii 4626	. 2 ⊢ {⟨y, x⟩ ∣ x(A ∘ B)y} = {⟨y, x⟩ ∣ ∃z(y◡Az ∧ z◡Bx)}
10		df-cnv 4785	. 2 ⊢ ◡(A ∘ B) = {⟨y, x⟩ ∣ x(A ∘ B)y}
11		df-co 4726	. 2 ⊢ (◡B ∘ ◡A) = {⟨y, x⟩ ∣ ∃z(y◡Az ∧ z◡Bx)}
12	9, 10, 11	3eqtr4i 2383	1 ⊢ ◡(A ∘ B) = (◡B ∘ ◡A)

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642  {copab 4622   class class class wbr 4639   ∘ ccom 4721  ^◡ccnv 4771

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-cnv 4785

This theorem is referenced by:  rncoss  4972  rncoeq  4975  dmco  5089  cores2  5091  coi2  5095  cnvtr  5098  f1co  5264  cnvpprod  5841  sbthlem3  6205