Theorem xrnss3v 38915

Description: A range Cartesian product is a subset of the class of ordered triples. This is Scott Fenton's txpss3v 36263 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 36263. (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 38914	. 2 ⊢ (𝐴 ⋉ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
2		inss1 4197	. . 3 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) ⊆ (◡(1st ↾ (V × V)) ∘ 𝐴)
3		relco 6108	. . . 4 ⊢ Rel (◡(1st ↾ (V × V)) ∘ 𝐴)
4		vex 3467	. . . . . . . . 9 ⊢ 𝑧 ∈ V
5		vex 3467	. . . . . . . . 9 ⊢ 𝑦 ∈ V
6	4, 5	brcnv 5866	. . . . . . . 8 ⊢ (𝑧◡(1st ↾ (V × V))𝑦 ↔ 𝑦(1st ↾ (V × V))𝑧)
7	4	brresi 5985	. . . . . . . . 9 ⊢ (𝑦(1st ↾ (V × V))𝑧 ↔ (𝑦 ∈ (V × V) ∧ 𝑦1st 𝑧))
8	7	simplbi 501	. . . . . . . 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 486	. . . . . 6 ⊢ ((𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V))
11	10	exlimiv 1957	. . . . 5 ⊢ (∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦) → 𝑦 ∈ (V × V))
12		vex 3467	. . . . . 6 ⊢ 𝑥 ∈ V
13	12, 5	opelco 5855	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (◡(1st ↾ (V × V)) ∘ 𝐴) ↔ ∃𝑧(𝑥𝐴𝑧 ∧ 𝑧◡(1st ↾ (V × V))𝑦))
14		opelxp 5695	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (V × (V × V)) ↔ (𝑥 ∈ V ∧ 𝑦 ∈ (V × V)))
15	12, 14	mpbiran 721	. . . . 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 5771	. . 3 ⊢ (◡(1st ↾ (V × V)) ∘ 𝐴) ⊆ (V × (V × V))
18	2, 17	sstri 3954	. 2 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) ⊆ (V × (V × V))
19	1, 18	eqsstri 3991	1 ⊢ (𝐴 ⋉ 𝐵) ⊆ (V × (V × V))

Syntax hints:   ∧ wa 400  ∃wex 1806   ∈ wcel 2149  Vcvv 3463   ∩ cin 3912   ⊆ wss 3913  ⟨cop 4597   class class class wbr 5110   × cxp 5657  ^◡ccnv 5658   ↾ cres 5661   ∘ ccom 5663  1^st c1st 7980  2^nd c2nd 7981   ⋉ cxrn 38708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-res 5671  df-xrn 38914

This theorem is referenced by:  xrnrel  38916  brxrn2  38918