Theorem dfco2 5234

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 5233	. 2 ⊢ Rel (𝐴 ∘ 𝐵)
2		reliun 4846	. . 3 ⊢ (Rel ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∀𝑥 ∈ V Rel ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
3		relxp 4833	. . . 4 ⊢ Rel ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
4	3	a1i 9	. . 3 ⊢ (𝑥 ∈ V → Rel ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
5	2, 4	mprgbir 2588	. 2 ⊢ Rel ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))
6		vex 2803	. . . 4 ⊢ 𝑦 ∈ V
7		vex 2803	. . . 4 ⊢ 𝑧 ∈ V
8		opelco2g 4896	. . . 4 ⊢ ((𝑦 ∈ V ∧ 𝑧 ∈ V) → (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴)))
9	6, 7, 8	mp2an 426	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
10		eliun 3972	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥 ∈ V ⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
11		rexv 2819	. . . 4 ⊢ (∃𝑥 ∈ V ⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
12		opelxp 4753	. . . . . 6 ⊢ (⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (𝑦 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑧 ∈ (𝐴 “ {𝑥})))
13		vex 2803	. . . . . . . . 9 ⊢ 𝑥 ∈ V
14	13, 6	elimasn 5101	. . . . . . . 8 ⊢ (𝑦 ∈ (◡𝐵 “ {𝑥}) ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝐵)
15	13, 6	opelcnv 4910	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡𝐵 ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
16	14, 15	bitri 184	. . . . . . 7 ⊢ (𝑦 ∈ (◡𝐵 “ {𝑥}) ↔ ⟨𝑦, 𝑥⟩ ∈ 𝐵)
17	13, 7	elimasn 5101	. . . . . . 7 ⊢ (𝑧 ∈ (𝐴 “ {𝑥}) ↔ ⟨𝑥, 𝑧⟩ ∈ 𝐴)
18	16, 17	anbi12i 460	. . . . . 6 ⊢ ((𝑦 ∈ (◡𝐵 “ {𝑥}) ∧ 𝑧 ∈ (𝐴 “ {𝑥})) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
19	12, 18	bitri 184	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ (⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
20	19	exbii 1651	. . . 4 ⊢ (∃𝑥⟨𝑦, 𝑧⟩ ∈ ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})) ↔ ∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴))
21	10, 11, 20	3bitrri 207	. . 3 ⊢ (∃𝑥(⟨𝑦, 𝑥⟩ ∈ 𝐵 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) ↔ ⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
22	9, 21	bitri 184	. 2 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥})))
23	1, 5, 22	eqrelriiv 4818	1 ⊢ (𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ V ((◡𝐵 “ {𝑥}) × (𝐴 “ {𝑥}))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∃wrex 2509  Vcvv 2800  {csn 3667  ⟨cop 3670  ∪ ciun 3968   × cxp 4721  ^◡ccnv 4722   “ cima 4726   ∘ ccom 4727  Rel wrel 4728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-iun 3970  df-br 4087  df-opab 4149  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736

This theorem is referenced by:  dfco2a  5235