Theorem dmpropg 6218

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 6216	. . 3 ⊢ (𝐵 ∈ 𝑉 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
2		dmsnopg 6216	. . 3 ⊢ (𝐷 ∈ 𝑊 → dom {⟨𝐶, 𝐷⟩} = {𝐶})
3		uneq12 4118	. . 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 4593	. . . 4 ⊢ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
6	5	dmeqi 5896	. . 3 ⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = dom ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
7		dmun 5902	. . 3 ⊢ dom ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (dom {⟨𝐴, 𝐵⟩} ∪ dom {⟨𝐶, 𝐷⟩})
8	6, 7	eqtri 2786	. 2 ⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = (dom {⟨𝐴, 𝐵⟩} ∪ dom {⟨𝐶, 𝐷⟩})
9		df-pr 4593	. 2 ⊢ {𝐴, 𝐶} = ({𝐴} ∪ {𝐶})
10	4, 8, 9	3eqtr4g 2823	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶})

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ∪ cun 3904  {csn 4590  {cpr 4592  ⟨cop 4596  dom cdm 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-dm 5673

This theorem is referenced by:  dmprop  6220  funtpg  6593  fnprg  6597  hashdmpropge2  14522  s2dmALT  14947  s4dom  14958  estrreslem2  18195  structiedg0val  29353