Theorem rnpropg 5184

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 3653	. . 3 ⊢ {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
2	1	rneqi 4928	. 2 ⊢ ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
3		rnsnopg 5183	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ran {⟨𝐴, 𝐶⟩} = {𝐶})
4	3	adantr 276	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran {⟨𝐴, 𝐶⟩} = {𝐶})
5		rnsnopg 5183	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → ran {⟨𝐵, 𝐷⟩} = {𝐷})
6	5	adantl 277	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran {⟨𝐵, 𝐷⟩} = {𝐷})
7	4, 6	uneq12d 3339	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩}) = ({𝐶} ∪ {𝐷}))
8		rnun 5113	. . 3 ⊢ ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}) = (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩})
9		df-pr 3653	. . 3 ⊢ {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
10	7, 8, 9	3eqtr4g 2267	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}) = {𝐶, 𝐷})
11	2, 10	eqtrid 2254	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {𝐶, 𝐷})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1375   ∈ wcel 2180   ∪ cun 3175  {csn 3646  {cpr 3647  ⟨cop 3649  ran crn 4697

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-br 4063  df-opab 4125  df-xp 4702  df-rel 4703  df-cnv 4704  df-dm 4706  df-rn 4707

This theorem is referenced by: (None)