Theorem brrangeg 35565

Description: Closed form of brrange 35563. (Contributed by Scott Fenton, 3-May-2014.)

Assertion

Ref	Expression
brrangeg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴))

Proof of Theorem brrangeg

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

Step	Hyp	Ref	Expression
1		breq1 5155	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎Range𝑏 ↔ 𝐴Range𝑏))
2		rneq 5942	. . . 4 ⊢ (𝑎 = 𝐴 → ran 𝑎 = ran 𝐴)
3	2	eqeq2d 2739	. . 3 ⊢ (𝑎 = 𝐴 → (𝑏 = ran 𝑎 ↔ 𝑏 = ran 𝐴))
4	1, 3	bibi12d 344	. 2 ⊢ (𝑎 = 𝐴 → ((𝑎Range𝑏 ↔ 𝑏 = ran 𝑎) ↔ (𝐴Range𝑏 ↔ 𝑏 = ran 𝐴)))
5		breq2 5156	. . 3 ⊢ (𝑏 = 𝐵 → (𝐴Range𝑏 ↔ 𝐴Range𝐵))
6		eqeq1 2732	. . 3 ⊢ (𝑏 = 𝐵 → (𝑏 = ran 𝐴 ↔ 𝐵 = ran 𝐴))
7	5, 6	bibi12d 344	. 2 ⊢ (𝑏 = 𝐵 → ((𝐴Range𝑏 ↔ 𝑏 = ran 𝐴) ↔ (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴)))
8		vex 3477	. . 3 ⊢ 𝑎 ∈ V
9		vex 3477	. . 3 ⊢ 𝑏 ∈ V
10	8, 9	brrange 35563	. 2 ⊢ (𝑎Range𝑏 ↔ 𝑏 = ran 𝑎)
11	4, 7, 10	vtocl2g 3562	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   class class class wbr 5152  ran crn 5683  Rangecrange 35473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-symdif 4245  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-eprel 5586  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fo 6559  df-fv 6561  df-1st 7999  df-2nd 8000  df-txp 35483  df-image 35493  df-range 35497

This theorem is referenced by: (None)