Theorem brdomaing 35917

Description: Closed form of brdomain 35915. (Contributed by Scott Fenton, 2-May-2014.)

Assertion

Ref	Expression
brdomaing	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴))

Proof of Theorem brdomaing

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 5151	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎Domain𝑏 ↔ 𝐴Domain𝑏))
2		dmeq 5917	. . . 4 ⊢ (𝑎 = 𝐴 → dom 𝑎 = dom 𝐴)
3	2	eqeq2d 2746	. . 3 ⊢ (𝑎 = 𝐴 → (𝑏 = dom 𝑎 ↔ 𝑏 = dom 𝐴))
4	1, 3	bibi12d 345	. 2 ⊢ (𝑎 = 𝐴 → ((𝑎Domain𝑏 ↔ 𝑏 = dom 𝑎) ↔ (𝐴Domain𝑏 ↔ 𝑏 = dom 𝐴)))
5		breq2 5152	. . 3 ⊢ (𝑏 = 𝐵 → (𝐴Domain𝑏 ↔ 𝐴Domain𝐵))
6		eqeq1 2739	. . 3 ⊢ (𝑏 = 𝐵 → (𝑏 = dom 𝐴 ↔ 𝐵 = dom 𝐴))
7	5, 6	bibi12d 345	. 2 ⊢ (𝑏 = 𝐵 → ((𝐴Domain𝑏 ↔ 𝑏 = dom 𝐴) ↔ (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴)))
8		vex 3482	. . 3 ⊢ 𝑎 ∈ V
9		vex 3482	. . 3 ⊢ 𝑏 ∈ V
10	8, 9	brdomain 35915	. 2 ⊢ (𝑎Domain𝑏 ↔ 𝑏 = dom 𝑎)
11	4, 7, 10	vtocl2g 3574	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106   class class class wbr 5148  dom cdm 5689  Domaincdomain 35825

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-symdif 4259  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-eprel 5589  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-1st 8013  df-2nd 8014  df-txp 35836  df-image 35846  df-domain 35849

This theorem is referenced by: (None)