Theorem dfco2 4840

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 4839	. 2 ⊢ Rel (𝐴 ∘ 𝐵)
2		reliun 4476	. . 3 ⊢ (Rel ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∀𝑥 ∈ V Rel ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
3		relxp 4465	. . . 4 ⊢ Rel ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
4	3	a1i 9	. . 3 ⊢ (𝑥 ∈ V → Rel ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
5	2, 4	mprgbir 2421	. 2 ⊢ Rel ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
6		vex 2604	. . . 4 ⊢ 𝑦 ∈ V
7		vex 2604	. . . 4 ⊢ 𝑧 ∈ V
8		opelco2g 4521	. . . 4 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴)))
9	6, 7, 8	mp2an 416	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
10		eliun 3682	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ V ⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
11		rexv 2617	. . . 4 ⊢ (∃𝑥 ∈ V ⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
12		opelxp 4392	. . . . . 6 ⊢ (⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (𝑦 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑧 ∈ (𝐴 “ {𝑥})))
13		vex 2604	. . . . . . . . 9 ⊢ 𝑥 ∈ V
14	13, 6	elimasn 4712	. . . . . . . 8 ⊢ (𝑦 ∈ (◡𝐵 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝐵)
15	13, 6	opelcnv 4535	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡𝐵 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
16	14, 15	bitri 182	. . . . . . 7 ⊢ (𝑦 ∈ (◡𝐵 “ {𝑥}) ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
17	13, 7	elimasn 4712	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐴)
18	16, 17	anbi12i 447	. . . . . 6 ⊢ ((𝑦 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑧 ∈ (𝐴 “ {𝑥})) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
19	12, 18	bitri 182	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
20	19	exbii 1536	. . . 4 ⊢ (∃𝑥⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
21	10, 11, 20	3bitrri 205	. . 3 ⊢ (∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) ↔ ⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
22	9, 21	bitri 182	. 2 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
23	1, 5, 22	eqrelriiv 4452	1 ⊢ (𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))

Syntax hints:   ∧ wa 102   ↔ wb 103   = wceq 1284  ∃wex 1421   ∈ wcel 1433  ∃wrex 2349  Vcvv 2601  {csn 3398  ⟨cop 3401  ∪ ciun 3678   × cxp 4361  ^◡ccnv 4362   “ cima 4366   ∘ ccom 4367  Rel wrel 4368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-iun 3680  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376

This theorem is referenced by:  dfco2a  4841