Theorem dfco2 5080

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 4884	. . 3 ⊢ (⟨y, z⟩ ∈ (A ∘ B) ↔ ∃x(yBx ∧ xAz))
2		eliun 3973	. . . 4 ⊢ (⟨y, z⟩ ∈ ∪x ∈ V ((◡B “ {x}) × (A “ {x})) ↔ ∃x ∈ V ⟨y, z⟩ ∈ ((◡B “ {x}) × (A “ {x})))
3		rexv 2873	. . . 4 ⊢ (∃x ∈ V ⟨y, z⟩ ∈ ((◡B “ {x}) × (A “ {x})) ↔ ∃x⟨y, z⟩ ∈ ((◡B “ {x}) × (A “ {x})))
4		opelxp 4811	. . . . . 6 ⊢ (⟨y, z⟩ ∈ ((◡B “ {x}) × (A “ {x})) ↔ (y ∈ (◡B “ {x}) ∧ z ∈ (A “ {x})))
5		eliniseg 5020	. . . . . . 7 ⊢ (y ∈ (◡B “ {x}) ↔ yBx)
6		elimasn 5019	. . . . . . . 8 ⊢ (z ∈ (A “ {x}) ↔ ⟨x, z⟩ ∈ A)
7		df-br 4640	. . . . . . . 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 4850	1 ⊢ (A ∘ B) = ∪x ∈ V ((◡B “ {x}) × (A “ {x}))

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  Vcvv 2859  {csn 3737  ∪ciun 3969  ⟨cop 4561   class class class wbr 4639   ∘ ccom 4721   “ cima 4722   × cxp 4770  ^◡ccnv 4771

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 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-iun 3971  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-ima 4727  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788

This theorem is referenced by:  dfco2a  5081