Theorem xpiun 41766

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 41765	. . . . 5 ⊢ ({𝑥} × 𝐶) = {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)}
2	1	eqcomi 2631	. . . 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 4539	. 2 ⊢ ∪ 𝑥 ∈ 𝐵 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)} = ∪ 𝑥 ∈ 𝐵 ({𝑥} × 𝐶)
5		iunxpconst 5175	. 2 ⊢ ∪ 𝑥 ∈ 𝐵 ({𝑥} × 𝐶) = (𝐵 × 𝐶)
6	4, 5	eqtr2i 2645	1 ⊢ (𝐵 × 𝐶) = ∪ 𝑥 ∈ 𝐵 {⟨𝑎, 𝑏⟩ ∣ (𝑎 = 𝑥 ∧ 𝑏 ∈ 𝐶)}

Syntax hints:   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {csn 4177  ∪ ciun 4520  {copab 4712   × cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-iun 4522  df-opab 4713  df-xp 5120

This theorem is referenced by: (None)