Theorem brrangeg 36248

Description: Closed form of brrange 36246. (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 5102	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎Range𝑏 ↔ 𝐴Range𝑏))
2		rneq 5910	. . . 4 ⊢ (𝑎 = 𝐴 → ran 𝑎 = ran 𝐴)
3	2	eqeq2d 2772	. . 3 ⊢ (𝑎 = 𝐴 → (𝑏 = ran 𝑎 ↔ 𝑏 = ran 𝐴))
4	1, 3	bibi12d 347	. 2 ⊢ (𝑎 = 𝐴 → ((𝑎Range𝑏 ↔ 𝑏 = ran 𝑎) ↔ (𝐴Range𝑏 ↔ 𝑏 = ran 𝐴)))
5		breq2 5103	. . 3 ⊢ (𝑏 = 𝐵 → (𝐴Range𝑏 ↔ 𝐴Range𝐵))
6		eqeq1 2765	. . 3 ⊢ (𝑏 = 𝐵 → (𝑏 = ran 𝐴 ↔ 𝐵 = ran 𝐴))
7	5, 6	bibi12d 347	. 2 ⊢ (𝑏 = 𝐵 → ((𝐴Range𝑏 ↔ 𝑏 = ran 𝐴) ↔ (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴)))
8		vex 3457	. . 3 ⊢ 𝑎 ∈ V
9		vex 3457	. . 3 ⊢ 𝑏 ∈ V
10	8, 9	brrange 36246	. 2 ⊢ (𝑎Range𝑏 ↔ 𝑏 = ran 𝑎)
11	4, 7, 10	vtocl2g 3538	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Range𝐵 ↔ 𝐵 = ran 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141   class class class wbr 5099  ran crn 5646  Rangecrange 36156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-symdif 4205  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-eprel 5545  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-fo 6523  df-fv 6525  df-1st 7966  df-2nd 7967  df-txp 36166  df-image 36176  df-range 36180

This theorem is referenced by: (None)