Theorem dfco2 6247

Description: Alternate definition of a class composition, using only one bound variable. (Contributed by NM, 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		relco 6111	. 2 ⊢ Rel (𝐴 ∘ 𝐵)
2		reliun 5804	. . 3 ⊢ (Rel ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∀𝑥 ∈ V Rel ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
3		relxp 5680	. . . 4 ⊢ Rel ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
4	3	a1i 11	. . 3 ⊢ (𝑥 ∈ V → Rel ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
5	2, 4	mprgbir 3092	. 2 ⊢ Rel ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
6		opelco2g 5854	. . . 4 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴)))
7	6	el2v 3470	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
8		eliun 4964	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ V ⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
9		rexv 3490	. . . 4 ⊢ (∃𝑥 ∈ V ⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
10		opelxp 5698	. . . . . 6 ⊢ (⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (𝑦 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑧 ∈ (𝐴 “ {𝑥})))
11		vex 3467	. . . . . . . . 9 ⊢ 𝑥 ∈ V
12		vex 3467	. . . . . . . . 9 ⊢ 𝑦 ∈ V
13	11, 12	elimasn 6093	. . . . . . . 8 ⊢ (𝑦 ∈ (◡𝐵 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝐵)
14	11, 12	opelcnv 5868	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡𝐵 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
15	13, 14	bitri 278	. . . . . . 7 ⊢ (𝑦 ∈ (◡𝐵 “ {𝑥}) ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
16		vex 3467	. . . . . . . 8 ⊢ 𝑧 ∈ V
17	11, 16	elimasn 6093	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐴)
18	15, 17	anbi12i 639	. . . . . 6 ⊢ ((𝑦 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑧 ∈ (𝐴 “ {𝑥})) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
19	10, 18	bitri 278	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
20	19	exbii 1875	. . . 4 ⊢ (∃𝑥⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
21	8, 9, 20	3bitrri 301	. . 3 ⊢ (∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) ↔ ⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
22	7, 21	bitri 278	. 2 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
23	1, 5, 22	eqrelriiv 5777	1 ⊢ (𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∃wrex 3095  Vcvv 3463  {csn 4594  ⟨cop 4600  ∪ ciun 4960   × cxp 5660  ^◡ccnv 5661   “ cima 5665   ∘ ccom 5666  Rel wrel 5667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-11 2198  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-iun 4962  df-br 5114  df-opab 5178  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675

This theorem is referenced by:  dfco2a  6248