Theorem brdomaing 36425

Description: Closed form of brdomain 36423. (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 5112	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎Domain𝑏 ↔ 𝐴Domain𝑏))
2		dmeq 5893	. . . 4 ⊢ (𝑎 = 𝐴 → dom 𝑎 = dom 𝐴)
3	2	eqeq2d 2774	. . 3 ⊢ (𝑎 = 𝐴 → (𝑏 = dom 𝑎 ↔ 𝑏 = dom 𝐴))
4	1, 3	bibi12d 348	. 2 ⊢ (𝑎 = 𝐴 → ((𝑎Domain𝑏 ↔ 𝑏 = dom 𝑎) ↔ (𝐴Domain𝑏 ↔ 𝑏 = dom 𝐴)))
5		breq2 5113	. . 3 ⊢ (𝑏 = 𝐵 → (𝐴Domain𝑏 ↔ 𝐴Domain𝐵))
6		eqeq1 2767	. . 3 ⊢ (𝑏 = 𝐵 → (𝑏 = dom 𝐴 ↔ 𝐵 = dom 𝐴))
7	5, 6	bibi12d 348	. 2 ⊢ (𝑏 = 𝐵 → ((𝐴Domain𝑏 ↔ 𝑏 = dom 𝐴) ↔ (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴)))
8		vex 3459	. . 3 ⊢ 𝑎 ∈ V
9		vex 3459	. . 3 ⊢ 𝑏 ∈ V
10	8, 9	brdomain 36423	. 2 ⊢ (𝑎Domain𝑏 ↔ 𝑏 = dom 𝑎)
11	4, 7, 10	vtocl2g 3538	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Domain𝐵 ↔ 𝐵 = dom 𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   class class class wbr 5109  dom cdm 5661  Domaincdomain 36333

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-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

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-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-symdif 4206  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-eprel 5561  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-1st 7982  df-2nd 7983  df-txp 36344  df-image 36354  df-domain 36357

This theorem is referenced by: (None)