Theorem brimageg 36275

Description: Closed form of brimage 36274. (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 5103	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥Image𝑅𝑦 ↔ 𝐴Image𝑅𝑦))
2		imaeq2 6045	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑅 “ 𝑥) = (𝑅 “ 𝐴))
3	2	eqeq2d 2773	. . 3 ⊢ (𝑥 = 𝐴 → (𝑦 = (𝑅 “ 𝑥) ↔ 𝑦 = (𝑅 “ 𝐴)))
4	1, 3	bibi12d 347	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥Image𝑅𝑦 ↔ 𝑦 = (𝑅 “ 𝑥)) ↔ (𝐴Image𝑅𝑦 ↔ 𝑦 = (𝑅 “ 𝐴))))
5		breq2 5104	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴Image𝑅𝑦 ↔ 𝐴Image𝑅𝐵))
6		eqeq1 2766	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 = (𝑅 “ 𝐴) ↔ 𝐵 = (𝑅 “ 𝐴)))
7	5, 6	bibi12d 347	. 2 ⊢ (𝑦 = 𝐵 → ((𝐴Image𝑅𝑦 ↔ 𝑦 = (𝑅 “ 𝐴)) ↔ (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴))))
8		vex 3458	. . 3 ⊢ 𝑥 ∈ V
9		vex 3458	. . 3 ⊢ 𝑦 ∈ V
10	8, 9	brimage 36274	. 2 ⊢ (𝑥Image𝑅𝑦 ↔ 𝑦 = (𝑅 “ 𝑥))
11	4, 7, 10	vtocl2g 3538	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142   class class class wbr 5100   “ cima 5650  Imagecimage 36188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-symdif 4205  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-eprel 5547  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-fo 6527  df-fv 6529  df-1st 7970  df-2nd 7971  df-txp 36202  df-image 36212

This theorem is referenced by:  fnimage  36277