Theorem xpiun 48906

Description: A Cartesian product expressed as indexed union of ordered-pair class abstractions. (Contributed by AV, 27-Jan-2020.)

Assertion

Ref	Expression
xpiun	⊢ (𝐵 × 𝐶) = ∪ 𝑥 ∈ 𝐵 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)}

Distinct variable groups:   𝑥,𝐵   𝐶,𝑎,𝑏,𝑥

Allowed substitution hints:   𝐵(𝑎,𝑏)

Proof of Theorem xpiun

Step	Hyp	Ref	Expression
1		xpsnopab 48905	. . . . 5 ⊢ ({𝑥} × 𝐶) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)}
2	1	eqcomi 2772	. . . 4 ⊢ {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)} = ({𝑥} × 𝐶)
3	2	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐵 → {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)} = ({𝑥} × 𝐶))
4	3	iuneq2i 4979	. 2 ⊢ ∪ 𝑥 ∈ 𝐵 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)} = ∪ 𝑥 ∈ 𝐵 ({𝑥} × 𝐶)
5		iunxpconst 5736	. 2 ⊢ ∪ 𝑥 ∈ 𝐵 ({𝑥} × 𝐶) = (𝐵 × 𝐶)
6	4, 5	eqtr2i 2787	1 ⊢ (𝐵 × 𝐶) = ∪ 𝑥 ∈ 𝐵 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)}

Syntax hints:   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {csn 4590  ∪ ciun 4957  {copab 5174   × cxp 5661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-11 2192  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-un 3911  df-in 3913  df-ss 3923  df-sn 4591  df-pr 4593  df-op 4597  df-iun 4959  df-opab 5175  df-xp 5669

This theorem is referenced by: (None)