Theorem rnpropg 6223

Description: The range of a pair of ordered pairs is the pair of second members. (Contributed by Thierry Arnoux, 3-Jan-2017.)

Assertion

Ref	Expression
rnpropg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {𝐶, 𝐷})

Proof of Theorem rnpropg

Step	Hyp	Ref	Expression
1		df-pr 4591	. . 3 ⊢ {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
2	1	rneqi 5927	. 2 ⊢ ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
3		rnsnopg 6222	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ran {⟨𝐴, 𝐶⟩} = {𝐶})
4	3	adantr 485	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran {⟨𝐴, 𝐶⟩} = {𝐶})
5		rnsnopg 6222	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → ran {⟨𝐵, 𝐷⟩} = {𝐷})
6	5	adantl 486	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran {⟨𝐵, 𝐷⟩} = {𝐷})
7	4, 6	uneq12d 4122	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩}) = ({𝐶} ∪ {𝐷}))
8		rnun 6142	. . 3 ⊢ ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}) = (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩})
9		df-pr 4591	. . 3 ⊢ {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
10	7, 8, 9	3eqtr4g 2821	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}) = {𝐶, 𝐷})
11	2, 10	eqtrid 2808	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {𝐶, 𝐷})

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568   ∈ wcel 2141   ∪ cun 3902  {csn 4588  {cpr 4590  ⟨cop 4594  ran crn 5662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672

This theorem is referenced by:  funcnvtp  6599  funcnvqp  6600  umgr2v2eedg  29840  esumsnf  34420  poimirlem9  38246  sge0sn  47063