Theorem brimageg 35203

Description: Closed form of brimage 35202. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
brimageg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴)))

Proof of Theorem brimageg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 5150	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥Image𝑅𝑦 ↔ 𝐴Image𝑅𝑦))
2		imaeq2 6054	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑅 “ 𝑥) = (𝑅 “ 𝐴))
3	2	eqeq2d 2741	. . 3 ⊢ (𝑥 = 𝐴 → (𝑦 = (𝑅 “ 𝑥) ↔ 𝑦 = (𝑅 “ 𝐴)))
4	1, 3	bibi12d 344	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥Image𝑅𝑦 ↔ 𝑦 = (𝑅 “ 𝑥)) ↔ (𝐴Image𝑅𝑦 ↔ 𝑦 = (𝑅 “ 𝐴))))
5		breq2 5151	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴Image𝑅𝑦 ↔ 𝐴Image𝑅𝐵))
6		eqeq1 2734	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 = (𝑅 “ 𝐴) ↔ 𝐵 = (𝑅 “ 𝐴)))
7	5, 6	bibi12d 344	. 2 ⊢ (𝑦 = 𝐵 → ((𝐴Image𝑅𝑦 ↔ 𝑦 = (𝑅 “ 𝐴)) ↔ (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴))))
8		vex 3476	. . 3 ⊢ 𝑥 ∈ V
9		vex 3476	. . 3 ⊢ 𝑦 ∈ V
10	8, 9	brimage 35202	. 2 ⊢ (𝑥Image𝑅𝑦 ↔ 𝑦 = (𝑅 “ 𝑥))
11	4, 7, 10	vtocl2g 3562	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104   class class class wbr 5147   “ cima 5678  Imagecimage 35116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-symdif 4241  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-eprel 5579  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550  df-1st 7977  df-2nd 7978  df-txp 35130  df-image 35140

This theorem is referenced by:  fnimage  35205