Theorem rnpropg 5241

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 3695	. . 3 ⊢ {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
2	1	rneqi 4984	. 2 ⊢ ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
3		rnsnopg 5240	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ran {⟨𝐴, 𝐶⟩} = {𝐶})
4	3	adantr 276	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran {⟨𝐴, 𝐶⟩} = {𝐶})
5		rnsnopg 5240	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → ran {⟨𝐵, 𝐷⟩} = {𝐷})
6	5	adantl 277	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran {⟨𝐵, 𝐷⟩} = {𝐷})
7	4, 6	uneq12d 3373	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩}) = ({𝐶} ∪ {𝐷}))
8		rnun 5170	. . 3 ⊢ ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}) = (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩})
9		df-pr 3695	. . 3 ⊢ {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
10	7, 8, 9	3eqtr4g 2290	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}) = {𝐶, 𝐷})
11	2, 10	eqtrid 2277	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {𝐶, 𝐷})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203   ∪ cun 3208  {csn 3688  {cpr 3689  ⟨cop 3691  ran crn 4749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109  df-opab 4171  df-xp 4754  df-rel 4755  df-cnv 4756  df-dm 4758  df-rn 4759

This theorem is referenced by: (None)