Theorem brimageg 33388

Description: Closed form of brimage 33387. (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 5069	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥Image𝑅𝑦 ↔ 𝐴Image𝑅𝑦))
2		imaeq2 5925	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑅 “ 𝑥) = (𝑅 “ 𝐴))
3	2	eqeq2d 2832	. . 3 ⊢ (𝑥 = 𝐴 → (𝑦 = (𝑅 “ 𝑥) ↔ 𝑦 = (𝑅 “ 𝐴)))
4	1, 3	bibi12d 348	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥Image𝑅𝑦 ↔ 𝑦 = (𝑅 “ 𝑥)) ↔ (𝐴Image𝑅𝑦 ↔ 𝑦 = (𝑅 “ 𝐴))))
5		breq2 5070	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴Image𝑅𝑦 ↔ 𝐴Image𝑅𝐵))
6		eqeq1 2825	. . 3 ⊢ (𝑦 = 𝐵 → (𝑦 = (𝑅 “ 𝐴) ↔ 𝐵 = (𝑅 “ 𝐴)))
7	5, 6	bibi12d 348	. 2 ⊢ (𝑦 = 𝐵 → ((𝐴Image𝑅𝑦 ↔ 𝑦 = (𝑅 “ 𝐴)) ↔ (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴))))
8		vex 3497	. . 3 ⊢ 𝑥 ∈ V
9		vex 3497	. . 3 ⊢ 𝑦 ∈ V
10	8, 9	brimage 33387	. 2 ⊢ (𝑥Image𝑅𝑦 ↔ 𝑦 = (𝑅 “ 𝑥))
11	4, 7, 10	vtocl2g 3572	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   class class class wbr 5066   “ cima 5558  Imagecimage 33301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-symdif 4219  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-eprel 5465  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fo 6361  df-fv 6363  df-1st 7689  df-2nd 7690  df-txp 33315  df-image 33325

This theorem is referenced by:  fnimage  33390