Theorem brimage 33965

Description: Binary relation form of the Image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Hypotheses

Ref	Expression
brimage.1	⊢ 𝐴 ∈ V
brimage.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
brimage	⊢ (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴))

Proof of Theorem brimage

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

Step	Hyp	Ref	Expression
1		brimage.1	. 2 ⊢ 𝐴 ∈ V
2		brimage.2	. 2 ⊢ 𝐵 ∈ V
3		df-image 33903	. 2 ⊢ Image𝑅 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝑅) ⊗ V)))
4		brxp 5598	. . 3 ⊢ (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
5	1, 2, 4	mpbir2an 711	. 2 ⊢ 𝐴(V × V)𝐵
6		vex 3412	. . . . 5 ⊢ 𝑥 ∈ V
7		vex 3412	. . . . 5 ⊢ 𝑦 ∈ V
8	6, 7	brcnv 5751	. . . 4 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
9	8	rexbii 3170	. . 3 ⊢ (∃𝑦 ∈ 𝐴 𝑥◡𝑅𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑦𝑅𝑥)
10	6, 1	coep 33437	. . 3 ⊢ (𝑥( E ∘ ◡𝑅)𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥◡𝑅𝑦)
11	6	elima 5934	. . 3 ⊢ (𝑥 ∈ (𝑅 “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑦𝑅𝑥)
12	9, 10, 11	3bitr4ri 307	. 2 ⊢ (𝑥 ∈ (𝑅 “ 𝐴) ↔ 𝑥( E ∘ ◡𝑅)𝐴)
13	1, 2, 3, 5, 12	brtxpsd3 33935	1 ⊢ (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴))

Syntax hints:   ↔ wb 209   = wceq 1543   ∈ wcel 2110  ∃wrex 3062  Vcvv 3408   class class class wbr 5053   E cep 5459   × cxp 5549  ^◡ccnv 5550   “ cima 5554   ∘ ccom 5555  Imagecimage 33879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-symdif 4157  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-eprel 5460  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fo 6386  df-fv 6388  df-1st 7761  df-2nd 7762  df-txp 33893  df-image 33903

This theorem is referenced by:  brimageg  33966  funimage  33967  fnimage  33968  imageval  33969  brdomain  33972  brrange  33973  brimg  33976  funpartlem  33981  imagesset  33992