Theorem xrnss3v 39572

Description: A range Cartesian product is a subset of the class of ordered triples. This is Scott Fenton's txpss3v 36848 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 36848. (Contributed by Scott Fenton, 31-Mar-2012.)

Assertion

Ref	Expression
xrnss3v	⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V))

Proof of Theorem xrnss3v

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-xrn 39571	. 2 ⊢ (𝐴 ⋉ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
2		inss1 4136	. . 3 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) ⊆ (◡(1st ↾ (V × V)) ∘ 𝐴)
3		relco 6120	. . . 4 ⊢ Rel (◡(1st ↾ (V × V)) ∘ 𝐴)
4		vex 3458	. . . . . . . . 9 ⊢ 𝑧 ∈ V
5		vex 3458	. . . . . . . . 9 ⊢ 𝑦 ∈ V
6	4, 5	brcnv 5870	. . . . . . . 8 ⊢ (𝑧◡(1st ↾ (V × V))𝑦 ↔ 𝑦(1st ↾ (V × V))𝑧)
7	4	brresi 5993	. . . . . . . . 9 ⊢ (𝑦(1st ↾ (V × V))𝑧 ↔ (𝑦 ∈ (V × V) ∧ 𝑦1st 𝑧))
8	7	simplbi 502	. . . . . . . 8 ⊢ (𝑦(1st ↾ (V × V))𝑧 → 𝑦 ∈ (V × V))
9	6, 8	sylbi 220	. . . . . . 7 ⊢ (𝑧◡(1st ↾ (V × V))𝑦 → 𝑦 ∈ (V × V))
10	9	adantl 487	. . . . . 6 ⊢ ((𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V))
11	10	exlimiv 1965	. . . . 5 ⊢ (∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V))
12		vex 3458	. . . . . 6 ⊢ 𝑥 ∈ V
13	12, 5	opelco 5859	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡(1st ↾ (V × V)) ∘ 𝐴) ↔ ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦))
14		opelxp 5692	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (V × (V × V)) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ (V × V)))
15	12, 14	mpbiran 722	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (V × (V × V)) ↔ 𝑦 ∈ (V × V))
16	11, 13, 15	3imtr4i 295	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡(1st ↾ (V × V)) ∘ 𝐴) → ⟨𝑥, 𝑦⟩ ∈ (V × (V × V)))
17	3, 16	relssi 5770	. . 3 ⊢ (◡(1st ↾ (V × V)) ∘ 𝐴) ⊆ (V × (V × V))
18	2, 17	sstri 3922	. 2 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) ⊆ (V × (V × V))
19	1, 18	eqsstri 3959	1 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V))

This proof depends on syntax axioms:   ∧ wa 401  ∃wex 1812   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  ⟨cop 4838   class class class wbr 5106   × cxp 5654  ^◡ccnv 5655   ↾ cres 5658   ∘ ccom 5660  1^st c1st 8031  2^nd c2nd 8032   ⋉ cxrn 39365

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

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-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-res 5668  df-xrn 39571

This theorem is used by:  xrnrel  39573  brxrn2  39575