Theorem rabxp 4815

Description: Membership in a class builder restricted to a cross product. (Contributed by NM, 20-Feb-2014.)

Hypothesis

Ref	Expression
rabxp.1	⊢ (x = ⟨y, z⟩ → (φ ↔ ψ))

Assertion

Ref	Expression
rabxp	⊢ {x ∈ (A × B) ∣ φ} = {⟨y, z⟩ ∣ (y ∈ A ∧ z ∈ B ∧ ψ)}

Distinct variable groups:   x,y,z,A   x,B,y,z   φ,y,z   ψ,x

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

Proof of Theorem rabxp

Step	Hyp	Ref	Expression
1		elxp 4802	. . . . 5 ⊢ (x ∈ (A × B) ↔ ∃y∃z(x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B)))
2	1	anbi1i 676	. . . 4 ⊢ ((x ∈ (A × B) ∧ φ) ↔ (∃y∃z(x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B)) ∧ φ))
3		19.41vv 1902	. . . 4 ⊢ (∃y∃z((x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B)) ∧ φ) ↔ (∃y∃z(x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B)) ∧ φ))
4		anass 630	. . . . . 6 ⊢ (((x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B)) ∧ φ) ↔ (x = ⟨y, z⟩ ∧ ((y ∈ A ∧ z ∈ B) ∧ φ)))
5		rabxp.1	. . . . . . . . 9 ⊢ (x = ⟨y, z⟩ → (φ ↔ ψ))
6	5	anbi2d 684	. . . . . . . 8 ⊢ (x = ⟨y, z⟩ → (((y ∈ A ∧ z ∈ B) ∧ φ) ↔ ((y ∈ A ∧ z ∈ B) ∧ ψ)))
7		df-3an 936	. . . . . . . 8 ⊢ ((y ∈ A ∧ z ∈ B ∧ ψ) ↔ ((y ∈ A ∧ z ∈ B) ∧ ψ))
8	6, 7	syl6bbr 254	. . . . . . 7 ⊢ (x = ⟨y, z⟩ → (((y ∈ A ∧ z ∈ B) ∧ φ) ↔ (y ∈ A ∧ z ∈ B ∧ ψ)))
9	8	pm5.32i 618	. . . . . 6 ⊢ ((x = ⟨y, z⟩ ∧ ((y ∈ A ∧ z ∈ B) ∧ φ)) ↔ (x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B ∧ ψ)))
10	4, 9	bitri 240	. . . . 5 ⊢ (((x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B)) ∧ φ) ↔ (x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B ∧ ψ)))
11	10	2exbii 1583	. . . 4 ⊢ (∃y∃z((x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B)) ∧ φ) ↔ ∃y∃z(x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B ∧ ψ)))
12	2, 3, 11	3bitr2i 264	. . 3 ⊢ ((x ∈ (A × B) ∧ φ) ↔ ∃y∃z(x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B ∧ ψ)))
13	12	abbii 2466	. 2 ⊢ {x ∣ (x ∈ (A × B) ∧ φ)} = {x ∣ ∃y∃z(x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B ∧ ψ))}
14		df-rab 2624	. 2 ⊢ {x ∈ (A × B) ∣ φ} = {x ∣ (x ∈ (A × B) ∧ φ)}
15		df-opab 4624	. 2 ⊢ {⟨y, z⟩ ∣ (y ∈ A ∧ z ∈ B ∧ ψ)} = {x ∣ ∃y∃z(x = ⟨y, z⟩ ∧ (y ∈ A ∧ z ∈ B ∧ ψ))}
16	13, 14, 15	3eqtr4i 2383	1 ⊢ {x ∈ (A × B) ∣ φ} = {⟨y, z⟩ ∣ (y ∈ A ∧ z ∈ B ∧ ψ)}

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2619  ⟨cop 4562  {copab 4623   × cxp 4771

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-opab 4624  df-xp 4785

This theorem is used by: (None)