Theorem brimageg 34229

Description: Closed form of brimage 34228. (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 5077	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥Image𝑅𝑦 ↔ 𝐴Image𝑅𝑦))
2		imaeq2 5965	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑅 “ 𝑥) = (𝑅 “ 𝐴))
3	2	eqeq2d 2749	. . 3 ⊢ (𝑥 = 𝐴 → (𝑦 = (𝑅 “ 𝑥) ↔ 𝑦 = (𝑅 “ 𝐴)))
4	1, 3	bibi12d 346	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥Image𝑅𝑦 ↔ 𝑦 = (𝑅 “ 𝑥)) ↔ (𝐴Image𝑅𝑦 ↔ 𝑦 = (𝑅 “ 𝐴))))
5		breq2 5078	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴Image𝑅𝑦 ↔ 𝐴Image𝑅𝐵))
6		eqeq1 2742	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 = (𝑅 “ 𝐴) ↔ 𝐵 = (𝑅 “ 𝐴)))
7	5, 6	bibi12d 346	. 2 ⊢ (𝑦 = 𝐵 → ((𝐴Image𝑅𝑦 ↔ 𝑦 = (𝑅 “ 𝐴)) ↔ (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴))))
8		vex 3436	. . 3 ⊢ 𝑥 ∈ V
9		vex 3436	. . 3 ⊢ 𝑦 ∈ V
10	8, 9	brimage 34228	. 2 ⊢ (𝑥Image𝑅𝑦 ↔ 𝑦 = (𝑅 “ 𝑥))
11	4, 7, 10	vtocl2g 3510	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   class class class wbr 5074   “ cima 5592  Imagecimage 34142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-symdif 4176  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-eprel 5495  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441  df-1st 7831  df-2nd 7832  df-txp 34156  df-image 34166

This theorem is referenced by:  fnimage  34231