Theorem xpab 36239

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 5684	. 2 ⊢ Rel ({𝑥 ∣ 𝜑} × {𝑦 ∣ 𝜓})
2		relopabv 5813	. 2 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}
3		df-clab 2745	. . . . 5 ⊢ (𝑎 ∈ {𝑥 ∣ 𝜑} ↔ [𝑎 / 𝑥]𝜑)
4		df-clab 2745	. . . . 5 ⊢ (𝑏 ∈ {𝑦 ∣ 𝜓} ↔ [𝑏 / 𝑦]𝜓)
5	3, 4	anbi12i 640	. . . 4 ⊢ ((𝑎 ∈ {𝑥 ∣ 𝜑} ∧ 𝑏 ∈ {𝑦 ∣ 𝜓}) ↔ ([𝑎 / 𝑥]𝜑 ∧ [𝑏 / 𝑦]𝜓))
6		sban 2117	. . . . . . 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 2125	. . . . . . . 8 ⊢ ([𝑏 / 𝑦]𝜑 ↔ 𝜑)
9	8	anbi1i 636	. . . . . . 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 2113	. . . . 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 2117	. . . . . 6 ⊢ ([𝑎 / 𝑥](𝜑 ∧ [𝑏 / 𝑦]𝜓) ↔ ([𝑎 / 𝑥]𝜑 ∧ [𝑎 / 𝑥][𝑏 / 𝑦]𝜓))
14		sbv 2125	. . . . . . 7 ⊢ ([𝑎 / 𝑥][𝑏 / 𝑦]𝜓 ↔ [𝑏 / 𝑦]𝜓)
15	14	anbi2i 635	. . . . . 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 5715	. . 3 ⊢ (𝑎({𝑥 ∣ 𝜑} × {𝑦 ∣ 𝜓})𝑏 ↔ (𝑎 ∈ {𝑥 ∣ 𝜑} ∧ 𝑏 ∈ {𝑦 ∣ 𝜓}))
20		eqid 2766	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)} = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}
21	20	brabsb 5520	. . 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 5782	1 ⊢ ({𝑥 ∣ 𝜑} × {𝑦 ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  [wsb 2099   ∈ wcel 2146  {cab 2744  [wsbc 3747   class class class wbr 5114  {copab 5178   × cxp 5664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-xp 5672  df-rel 5673

This theorem is used by: (None)