Theorem xpab 35688

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 5718	. 2 ⊢ Rel ({𝑥 ∣ 𝜑} × {𝑦 ∣ 𝜓})
2		relopabv 5845	. 2 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}
3		df-clab 2718	. . . . 5 ⊢ (𝑎 ∈ {𝑥 ∣ 𝜑} ↔ [𝑎 / 𝑥]𝜑)
4		df-clab 2718	. . . . 5 ⊢ (𝑏 ∈ {𝑦 ∣ 𝜓} ↔ [𝑏 / 𝑦]𝜓)
5	3, 4	anbi12i 627	. . . 4 ⊢ ((𝑎 ∈ {𝑥 ∣ 𝜑} ∧ 𝑏 ∈ {𝑦 ∣ 𝜓}) ↔ ([𝑎 / 𝑥]𝜑 ∧ [𝑏 / 𝑦]𝜓))
6		sban 2080	. . . . . . 7 ⊢ ([𝑏 / 𝑦](𝜑 ∧ 𝜓) ↔ ([𝑏 / 𝑦]𝜑 ∧ [𝑏 / 𝑦]𝜓))
7		sbsbc 3808	. . . . . . 7 ⊢ ([𝑏 / 𝑦](𝜑 ∧ 𝜓) ↔ [𝑏 / 𝑦](𝜑 ∧ 𝜓))
8		sbv 2088	. . . . . . . 8 ⊢ ([𝑏 / 𝑦]𝜑 ↔ 𝜑)
9	8	anbi1i 623	. . . . . . 7 ⊢ (([𝑏 / 𝑦]𝜑 ∧ [𝑏 / 𝑦]𝜓) ↔ (𝜑 ∧ [𝑏 / 𝑦]𝜓))
10	6, 7, 9	3bitr3i 301	. . . . . 6 ⊢ ([𝑏 / 𝑦](𝜑 ∧ 𝜓) ↔ (𝜑 ∧ [𝑏 / 𝑦]𝜓))
11	10	sbbii 2076	. . . . 5 ⊢ ([𝑎 / 𝑥][𝑏 / 𝑦](𝜑 ∧ 𝜓) ↔ [𝑎 / 𝑥](𝜑 ∧ [𝑏 / 𝑦]𝜓))
12		sbsbc 3808	. . . . 5 ⊢ ([𝑎 / 𝑥][𝑏 / 𝑦](𝜑 ∧ 𝜓) ↔ [𝑎 / 𝑥][𝑏 / 𝑦](𝜑 ∧ 𝜓))
13		sban 2080	. . . . . 6 ⊢ ([𝑎 / 𝑥](𝜑 ∧ [𝑏 / 𝑦]𝜓) ↔ ([𝑎 / 𝑥]𝜑 ∧ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓))
14		sbv 2088	. . . . . . 7 ⊢ ([𝑎 / 𝑥][𝑏 / 𝑦]𝜓 ↔ [𝑏 / 𝑦]𝜓)
15	14	anbi2i 622	. . . . . 6 ⊢ (([𝑎 / 𝑥]𝜑 ∧ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓) ↔ ([𝑎 / 𝑥]𝜑 ∧ [𝑏 / 𝑦]𝜓))
16	13, 15	bitri 275	. . . . 5 ⊢ ([𝑎 / 𝑥](𝜑 ∧ [𝑏 / 𝑦]𝜓) ↔ ([𝑎 / 𝑥]𝜑 ∧ [𝑏 / 𝑦]𝜓))
17	11, 12, 16	3bitr3i 301	. . . 4 ⊢ ([𝑎 / 𝑥][𝑏 / 𝑦](𝜑 ∧ 𝜓) ↔ ([𝑎 / 𝑥]𝜑 ∧ [𝑏 / 𝑦]𝜓))
18	5, 17	bitr4i 278	. . 3 ⊢ ((𝑎 ∈ {𝑥 ∣ 𝜑} ∧ 𝑏 ∈ {𝑦 ∣ 𝜓}) ↔ [𝑎 / 𝑥][𝑏 / 𝑦](𝜑 ∧ 𝜓))
19		brxp 5749	. . 3 ⊢ (𝑎({𝑥 ∣ 𝜑} × {𝑦 ∣ 𝜓})𝑏 ↔ (𝑎 ∈ {𝑥 ∣ 𝜑} ∧ 𝑏 ∈ {𝑦 ∣ 𝜓}))
20		eqid 2740	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)} = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}
21	20	brabsb 5550	. . 3 ⊢ (𝑎{⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}𝑏 ↔ [𝑎 / 𝑥][𝑏 / 𝑦](𝜑 ∧ 𝜓))
22	18, 19, 21	3bitr4i 303	. 2 ⊢ (𝑎({𝑥 ∣ 𝜑} × {𝑦 ∣ 𝜓})𝑏 ↔ 𝑎{⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}𝑏)
23	1, 2, 22	eqbrriv 5815	1 ⊢ ({𝑥 ∣ 𝜑} × {𝑦 ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}

Syntax hints:   ∧ wa 395   = wceq 1537  [wsb 2064   ∈ wcel 2108  {cab 2717  [wsbc 3804   class class class wbr 5166  {copab 5228   × cxp 5698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707

This theorem is referenced by: (None)