Theorem dfco2 6209

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 6073	. 2 ⊢ Rel (𝐴 ∘ 𝐵)
2		reliun 5772	. . 3 ⊢ (Rel ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∀𝑥 ∈ V Rel ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
3		relxp 5649	. . . 4 ⊢ Rel ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
4	3	a1i 11	. . 3 ⊢ (𝑥 ∈ V → Rel ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
5	2, 4	mprgbir 3058	. 2 ⊢ Rel ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
6		opelco2g 5822	. . . 4 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴)))
7	6	el2v 3436	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
8		eliun 4937	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ V ⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
9		rexv 3457	. . . 4 ⊢ (∃𝑥 ∈ V ⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
10		opelxp 5667	. . . . . 6 ⊢ (⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (𝑦 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑧 ∈ (𝐴 “ {𝑥})))
11		vex 3433	. . . . . . . . 9 ⊢ 𝑥 ∈ V
12		vex 3433	. . . . . . . . 9 ⊢ 𝑦 ∈ V
13	11, 12	elimasn 6055	. . . . . . . 8 ⊢ (𝑦 ∈ (◡𝐵 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝐵)
14	11, 12	opelcnv 5836	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡𝐵 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
15	13, 14	bitri 275	. . . . . . 7 ⊢ (𝑦 ∈ (◡𝐵 “ {𝑥}) ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
16		vex 3433	. . . . . . . 8 ⊢ 𝑧 ∈ V
17	11, 16	elimasn 6055	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐴)
18	15, 17	anbi12i 629	. . . . . 6 ⊢ ((𝑦 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑧 ∈ (𝐴 “ {𝑥})) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
19	10, 18	bitri 275	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
20	19	exbii 1850	. . . 4 ⊢ (∃𝑥⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
21	8, 9, 20	3bitrri 298	. . 3 ⊢ (∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) ↔ ⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
22	7, 21	bitri 275	. 2 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
23	1, 5, 22	eqrelriiv 5746	1 ⊢ (𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3061  Vcvv 3429  {csn 4567  ⟨cop 4573  ∪ ciun 4933   × cxp 5629  ^◡ccnv 5630   “ cima 5634   ∘ ccom 5635  Rel wrel 5636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-iun 4935  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644

This theorem is referenced by:  dfco2a  6210