Theorem dmpropg 6205

Description: The domain of an unordered pair of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.)

Assertion

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

Proof of Theorem dmpropg

Step	Hyp	Ref	Expression
1		dmsnopg 6203	. . 3 ⊢ (𝐵 ∈ 𝑉 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
2		dmsnopg 6203	. . 3 ⊢ (𝐷 ∈ 𝑊 → dom {⟨𝐶, 𝐷⟩} = {𝐶})
3		uneq12 4119	. . 3 ⊢ ((dom {⟨𝐴, 𝐵⟩} = {𝐴} ∧ dom {⟨𝐶, 𝐷⟩} = {𝐶}) → (dom {⟨𝐴, 𝐵⟩} ∪ dom {⟨𝐶, 𝐷⟩}) = ({𝐴} ∪ {𝐶}))
4	1, 2, 3	syl2an 607	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (dom {⟨𝐴, 𝐵⟩} ∪ dom {⟨𝐶, 𝐷⟩}) = ({𝐴} ∪ {𝐶}))
5		df-pr 4588	. . . 4 ⊢ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
6	5	dmeqi 5884	. . 3 ⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = dom ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
7		dmun 5890	. . 3 ⊢ dom ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (dom {⟨𝐴, 𝐵⟩} ∪ dom {⟨𝐶, 𝐷⟩})
8	6, 7	eqtri 2788	. 2 ⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = (dom {⟨𝐴, 𝐵⟩} ∪ dom {⟨𝐶, 𝐷⟩})
9		df-pr 4588	. 2 ⊢ {𝐴, 𝐶} = ({𝐴} ∪ {𝐶})
10	4, 8, 9	3eqtr4g 2825	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶})

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ∪ cun 3905  {csn 4585  {cpr 4587  ⟨cop 4591  dom cdm 5651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5250  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-dm 5661

This theorem is referenced by:  dmprop  6207  funtpg  6580  fnprg  6584  hashdmpropge2  14508  s2dmALT  14933  s4dom  14944  estrreslem2  18182  structiedg0val  29277