Theorem dfco2 6197

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 6060	. 2 ⊢ Rel (𝐴 ∘ 𝐵)
2		reliun 5772	. . 3 ⊢ (Rel ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∀𝑥 ∈ V Rel ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
3		relxp 5651	. . . 4 ⊢ Rel ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
4	3	a1i 11	. . 3 ⊢ (𝑥 ∈ V → Rel ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
5	2, 4	mprgbir 3071	. 2 ⊢ Rel ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
6		opelco2g 5823	. . . 4 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴)))
7	6	el2v 3453	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
8		eliun 4958	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ V ⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
9		rexv 3470	. . . 4 ⊢ (∃𝑥 ∈ V ⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
10		opelxp 5669	. . . . . 6 ⊢ (⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (𝑦 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑧 ∈ (𝐴 “ {𝑥})))
11		vex 3449	. . . . . . . . 9 ⊢ 𝑥 ∈ V
12		vex 3449	. . . . . . . . 9 ⊢ 𝑦 ∈ V
13	11, 12	elimasn 6041	. . . . . . . 8 ⊢ (𝑦 ∈ (◡𝐵 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝐵)
14	11, 12	opelcnv 5837	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡𝐵 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
15	13, 14	bitri 274	. . . . . . 7 ⊢ (𝑦 ∈ (◡𝐵 “ {𝑥}) ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
16		vex 3449	. . . . . . . 8 ⊢ 𝑧 ∈ V
17	11, 16	elimasn 6041	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐴)
18	15, 17	anbi12i 627	. . . . . 6 ⊢ ((𝑦 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑧 ∈ (𝐴 “ {𝑥})) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
19	10, 18	bitri 274	. . . . 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 297	. . 3 ⊢ (∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) ↔ ⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
22	7, 21	bitri 274	. 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 205   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∃wrex 3073  Vcvv 3445  {csn 4586  ⟨cop 4592  ∪ ciun 4954   × cxp 5631  ^◡ccnv 5632   “ cima 5636   ∘ ccom 5637  Rel wrel 5638

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-iun 4956  df-br 5106  df-opab 5168  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646

This theorem is referenced by:  dfco2a  6198