Theorem dfco2 5081

Description: Alternate definition of a class composition, using only one bound variable. (Contributed by set.mm contributors, 19-Dec-2008.)

Assertion

Ref	Expression
dfco2	⊢ (A ∘ B) = ∪x ∈ V ((◡B “ {x}) × (A “ {x}))

Distinct variable groups:   x,A   x,B

Proof of Theorem dfco2

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

Step	Hyp	Ref	Expression
1		opelco 4885	. . 3 ⊢ (⟨y, z⟩ ∈ (A ∘ B) ↔ ∃x(yBx ∧ xAz))
2		eliun 3974	. . . 4 ⊢ (⟨y, z⟩ ∈ ∪x ∈ V ((◡B “ {x}) × (A “ {x})) ↔ ∃x ∈ V ⟨y, z⟩ ∈ ((◡B “ {x}) × (A “ {x})))
3		rexv 2874	. . . 4 ⊢ (∃x ∈ V ⟨y, z⟩ ∈ ((◡B “ {x}) × (A “ {x})) ↔ ∃x⟨y, z⟩ ∈ ((◡B “ {x}) × (A “ {x})))
4		opelxp 4812	. . . . . 6 ⊢ (⟨y, z⟩ ∈ ((◡B “ {x}) × (A “ {x})) ↔ (y ∈ (◡B “ {x}) ∧ z ∈ (A “ {x})))
5		eliniseg 5021	. . . . . . 7 ⊢ (y ∈ (◡B “ {x}) ↔ yBx)
6		elimasn 5020	. . . . . . . 8 ⊢ (z ∈ (A “ {x}) ↔ ⟨x, z⟩ ∈ A)
7		df-br 4641	. . . . . . . 8 ⊢ (xAz ↔ ⟨x, z⟩ ∈ A)
8	6, 7	bitr4i 243	. . . . . . 7 ⊢ (z ∈ (A “ {x}) ↔ xAz)
9	5, 8	anbi12i 678	. . . . . 6 ⊢ ((y ∈ (◡B “ {x}) ∧ z ∈ (A “ {x})) ↔ (yBx ∧ xAz))
10	4, 9	bitri 240	. . . . 5 ⊢ (⟨y, z⟩ ∈ ((◡B “ {x}) × (A “ {x})) ↔ (yBx ∧ xAz))
11	10	exbii 1582	. . . 4 ⊢ (∃x⟨y, z⟩ ∈ ((◡B “ {x}) × (A “ {x})) ↔ ∃x(yBx ∧ xAz))
12	2, 3, 11	3bitrri 263	. . 3 ⊢ (∃x(yBx ∧ xAz) ↔ ⟨y, z⟩ ∈ ∪x ∈ V ((◡B “ {x}) × (A “ {x})))
13	1, 12	bitri 240	. 2 ⊢ (⟨y, z⟩ ∈ (A ∘ B) ↔ ⟨y, z⟩ ∈ ∪x ∈ V ((◡B “ {x}) × (A “ {x})))
14	13	eqrelriv 4851	1 ⊢ (A ∘ B) = ∪x ∈ V ((◡B “ {x}) × (A “ {x}))

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  Vcvv 2860  {csn 3738  ∪ciun 3970  ⟨cop 4562   class class class wbr 4640   ∘ ccom 4722   “ cima 4723   × cxp 4771  ^◡ccnv 4772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789

This theorem is referenced by:  dfco2a  5082