Theorem dmpropg 6173

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 6171	. . 3 ⊢ (𝐵 ∈ 𝑉 → dom {⟨𝐴, 𝐵⟩} = {𝐴})
2		dmsnopg 6171	. . 3 ⊢ (𝐷 ∈ 𝑊 → dom {⟨𝐶, 𝐷⟩} = {𝐶})
3		uneq12 4100	. . 3 ⊢ ((dom {⟨𝐴, 𝐵⟩} = {𝐴} ∧ dom {⟨𝐶, 𝐷⟩} = {𝐶}) → (dom {⟨𝐴, 𝐵⟩} ∪ dom {⟨𝐶, 𝐷⟩}) = ({𝐴} ∪ {𝐶}))
4	1, 2, 3	syl2an 602	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (dom {⟨𝐴, 𝐵⟩} ∪ dom {⟨𝐶, 𝐷⟩}) = ({𝐴} ∪ {𝐶}))
5		df-pr 4565	. . . 4 ⊢ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
6	5	dmeqi 5853	. . 3 ⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = dom ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
7		dmun 5859	. . 3 ⊢ dom ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (dom {⟨𝐴, 𝐵⟩} ∪ dom {⟨𝐶, 𝐷⟩})
8	6, 7	eqtri 2763	. 2 ⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = (dom {⟨𝐴, 𝐵⟩} ∪ dom {⟨𝐶, 𝐷⟩})
9		df-pr 4565	. 2 ⊢ {𝐴, 𝐶} = ({𝐴} ∪ {𝐶})
10	4, 8, 9	3eqtr4g 2800	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ∪ cun 3888  {csn 4562  {cpr 4564  ⟨cop 4568  dom cdm 5625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-dm 5635

This theorem is referenced by:  dmprop  6175  funtpg  6547  fnprg  6551  hashdmpropge2  14443  s2dmALT  14868  s4dom  14879  estrreslem2  18102  structiedg0val  29116