Theorem elco 4941

Description: Elements of a composed relation. (Contributed by BJ, 10-Jul-2022.)

Assertion

Ref	Expression
elco	⊢ (𝐴 ∈ (𝑅 ∘ 𝑆) ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)))

Distinct variable groups:   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝑥,𝐴,𝑦,𝑧

Proof of Theorem elco

Step	Hyp	Ref	Expression
1		df-co 4778	. . 3 ⊢ (𝑅 ∘ 𝑆) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)}
2	1	eleq2i 2305	. 2 ⊢ (𝐴 ∈ (𝑅 ∘ 𝑆) ↔ 𝐴 ∈ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)})
3		elopab 4395	. . 3 ⊢ (𝐴 ∈ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)} ↔ ∃𝑥∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)))
4		19.42v 1962	. . . . . . 7 ⊢ (∃𝑦(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)) ↔ (𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)))
5	4	bicomi 132	. . . . . 6 ⊢ ((𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)) ↔ ∃𝑦(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)))
6	5	exbii 1658	. . . . 5 ⊢ (∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)) ↔ ∃𝑧∃𝑦(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)))
7		excom 1716	. . . . 5 ⊢ (∃𝑧∃𝑦(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)) ↔ ∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)))
8	6, 7	bitri 184	. . . 4 ⊢ (∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)) ↔ ∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)))
9	8	exbii 1658	. . 3 ⊢ (∃𝑥∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)) ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)))
10	3, 9	bitri 184	. 2 ⊢ (𝐴 ∈ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)} ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)))
11	2, 10	bitri 184	1 ⊢ (𝐴 ∈ (𝑅 ∘ 𝑆) ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑥, 𝑧⟩ ∧ (𝑥𝑆𝑦 ∧ 𝑦𝑅𝑧)))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ⟨cop 3708   class class class wbr 4125  {copab 4186   ∘ ccom 4773

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-opab 4188  df-co 4778

This theorem is referenced by: (None)