Theorem xpab 36583

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 5668	. 2 ⊢ Rel ({𝑥 ∣ 𝜑} × {𝑦 ∣ 𝜓})
2		relopabv 5798	. 2 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}
3		df-clab 2739	. . . . 5 ⊢ (𝑎 ∈ {𝑥 ∣ 𝜑} ↔ [𝑎 / 𝑥]𝜑)
4		df-clab 2739	. . . . 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 2116	. . . . . . 7 ⊢ ([𝑏 / 𝑦](𝜑 ∧ 𝜓) ↔ ([𝑏 / 𝑦]𝜑 ∧ [𝑏 / 𝑦]𝜓))
7		sbsbc 3742	. . . . . . 7 ⊢ ([𝑏 / 𝑦](𝜑 ∧ 𝜓) ↔ [𝑏 / 𝑦](𝜑 ∧ 𝜓))
8		sbv 2124	. . . . . . . 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 2112	. . . . 5 ⊢ ([𝑎 / 𝑥][𝑏 / 𝑦](𝜑 ∧ 𝜓) ↔ [𝑎 / 𝑥](𝜑 ∧ [𝑏 / 𝑦]𝜓))
12		sbsbc 3742	. . . . 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 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 5699	. . 3 ⊢ (𝑎({𝑥 ∣ 𝜑} × {𝑦 ∣ 𝜓})𝑏 ↔ (𝑎 ∈ {𝑥 ∣ 𝜑} ∧ 𝑏 ∈ {𝑦 ∣ 𝜓}))
20		eqid 2760	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)} = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}
21	20	brabsb 5504	. . 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 5766	1 ⊢ ({𝑥 ∣ 𝜑} × {𝑦 ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569  [wsb 2098   ∈ wcel 2144  {cab 2738  [wsbc 3738   class class class wbr 5102  {copab 5166   × cxp 5648

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657

This theorem is used by: (None)