Theorem xpiun 48656

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 48655	. . . . 5 ⊢ ({𝑥} × 𝐶) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)}
2	1	eqcomi 2749	. . . 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 4950	. 2 ⊢ ∪ 𝑥 ∈ 𝐵 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)} = ∪ 𝑥 ∈ 𝐵 ({𝑥} × 𝐶)
5		iunxpconst 5698	. 2 ⊢ ∪ 𝑥 ∈ 𝐵 ({𝑥} × 𝐶) = (𝐵 × 𝐶)
6	4, 5	eqtr2i 2764	1 ⊢ (𝐵 × 𝐶) = ∪ 𝑥 ∈ 𝐵 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)}

Syntax hints:   ∧ wa 396   = wceq 1547   ∈ wcel 2119  {csn 4562  ∪ ciun 4928  {copab 5141   × cxp 5623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-11 2168  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-un 3895  df-in 3897  df-ss 3907  df-sn 4563  df-pr 4565  df-op 4569  df-iun 4930  df-opab 5142  df-xp 5631

This theorem is referenced by: (None)