Theorem xpab 36089

Description: Cartesian product of two class abstractions. (Contributed by Scott Fenton, 19-Aug-2024.)

Assertion

Ref	Expression
xpab	⊢ ({𝑥 ∣ 𝜑} × {𝑦 ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}

Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem xpab

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relxp 5670	. 2 ⊢ Rel ({𝑥 ∣ 𝜑} × {𝑦 ∣ 𝜓})
2		relopabv 5799	. 2 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}
3		df-clab 2744	. . . . 5 ⊢ (𝑎 ∈ {𝑥 ∣ 𝜑} ↔ [𝑎 / 𝑥]𝜑)
4		df-clab 2744	. . . . 5 ⊢ (𝑏 ∈ {𝑦 ∣ 𝜓} ↔ [𝑏 / 𝑦]𝜓)
5	3, 4	anbi12i 639	. . . 4 ⊢ ((𝑎 ∈ {𝑥 ∣ 𝜑} ∧ 𝑏 ∈ {𝑦 ∣ 𝜓}) ↔ ([𝑎 / 𝑥]𝜑 ∧ [𝑏 / 𝑦]𝜓))
6		sban 2116	. . . . . . 7 ⊢ ([𝑏 / 𝑦](𝜑 ∧ 𝜓) ↔ ([𝑏 / 𝑦]𝜑 ∧ [𝑏 / 𝑦]𝜓))
7		sbsbc 3751	. . . . . . 7 ⊢ ([𝑏 / 𝑦](𝜑 ∧ 𝜓) ↔ [𝑏 / 𝑦](𝜑 ∧ 𝜓))
8		sbv 2124	. . . . . . . 8 ⊢ ([𝑏 / 𝑦]𝜑 ↔ 𝜑)
9	8	anbi1i 635	. . . . . . 7 ⊢ (([𝑏 / 𝑦]𝜑 ∧ [𝑏 / 𝑦]𝜓) ↔ (𝜑 ∧ [𝑏 / 𝑦]𝜓))
10	6, 7, 9	3bitr3i 304	. . . . . 6 ⊢ ([𝑏 / 𝑦](𝜑 ∧ 𝜓) ↔ (𝜑 ∧ [𝑏 / 𝑦]𝜓))
11	10	sbbii 2112	. . . . 5 ⊢ ([𝑎 / 𝑥][𝑏 / 𝑦](𝜑 ∧ 𝜓) ↔ [𝑎 / 𝑥](𝜑 ∧ [𝑏 / 𝑦]𝜓))
12		sbsbc 3751	. . . . 5 ⊢ ([𝑎 / 𝑥][𝑏 / 𝑦](𝜑 ∧ 𝜓) ↔ [𝑎 / 𝑥][𝑏 / 𝑦](𝜑 ∧ 𝜓))
13		sban 2116	. . . . . 6 ⊢ ([𝑎 / 𝑥](𝜑 ∧ [𝑏 / 𝑦]𝜓) ↔ ([𝑎 / 𝑥]𝜑 ∧ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓))
14		sbv 2124	. . . . . . 7 ⊢ ([𝑎 / 𝑥][𝑏 / 𝑦]𝜓 ↔ [𝑏 / 𝑦]𝜓)
15	14	anbi2i 634	. . . . . 6 ⊢ (([𝑎 / 𝑥]𝜑 ∧ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓) ↔ ([𝑎 / 𝑥]𝜑 ∧ [𝑏 / 𝑦]𝜓))
16	13, 15	bitri 278	. . . . 5 ⊢ ([𝑎 / 𝑥](𝜑 ∧ [𝑏 / 𝑦]𝜓) ↔ ([𝑎 / 𝑥]𝜑 ∧ [𝑏 / 𝑦]𝜓))
17	11, 12, 16	3bitr3i 304	. . . 4 ⊢ ([𝑎 / 𝑥][𝑏 / 𝑦](𝜑 ∧ 𝜓) ↔ ([𝑎 / 𝑥]𝜑 ∧ [𝑏 / 𝑦]𝜓))
18	5, 17	bitr4i 281	. . 3 ⊢ ((𝑎 ∈ {𝑥 ∣ 𝜑} ∧ 𝑏 ∈ {𝑦 ∣ 𝜓}) ↔ [𝑎 / 𝑥][𝑏 / 𝑦](𝜑 ∧ 𝜓))
19		brxp 5701	. . 3 ⊢ (𝑎({𝑥 ∣ 𝜑} × {𝑦 ∣ 𝜓})𝑏 ↔ (𝑎 ∈ {𝑥 ∣ 𝜑} ∧ 𝑏 ∈ {𝑦 ∣ 𝜓}))
20		eqid 2765	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)} = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}
21	20	brabsb 5506	. . 3 ⊢ (𝑎{⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}𝑏 ↔ [𝑎 / 𝑥][𝑏 / 𝑦](𝜑 ∧ 𝜓))
22	18, 19, 21	3bitr4i 306	. 2 ⊢ (𝑎({𝑥 ∣ 𝜑} × {𝑦 ∣ 𝜓})𝑏 ↔ 𝑎{⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}𝑏)
23	1, 2, 22	eqbrriv 5768	1 ⊢ ({𝑥 ∣ 𝜑} × {𝑦 ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}

Syntax hints:   ∧ wa 400   = wceq 1563  [wsb 2093   ∈ wcel 2145  {cab 2743  [wsbc 3747   class class class wbr 5105  {copab 5167   × cxp 5650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-rel 5659

This theorem is referenced by: (None)