Theorem brimage 36315

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 36253	. 2 ⊢ Image𝑅 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝑅) ⊗ V)))
4		brxp 5711	. . 3 ⊢ (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
5	1, 2, 4	mpbir2an 723	. 2 ⊢ 𝐴(V × V)𝐵
6		vex 3467	. . . . 5 ⊢ 𝑥 ∈ V
7		vex 3467	. . . . 5 ⊢ 𝑦 ∈ V
8	6, 7	brcnv 5869	. . . 4 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
9	8	rexbii 3118	. . 3 ⊢ (∃𝑦 ∈ 𝐴 𝑥◡𝑅𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑦𝑅𝑥)
10	6, 1	coep 36143	. . 3 ⊢ (𝑥( E ∘ ◡𝑅)𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥◡𝑅𝑦)
11	6	elima 6068	. . 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 36285	1 ⊢ (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴))

Syntax hints:   ↔ wb 209   = wceq 1567   ∈ wcel 2149  ∃wrex 3095  Vcvv 3463   class class class wbr 5113   E cep 5561   × cxp 5660  ^◡ccnv 5661   “ cima 5665   ∘ ccom 5666  Imagecimage 36229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-symdif 4214  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-eprel 5562  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-1st 7986  df-2nd 7987  df-txp 36243  df-image 36253

This theorem is referenced by:  brimageg  36316  funimage  36317  fnimage  36318  imageval  36319  brdomain  36322  brrange  36323  brimg  36326  funpartlem  36333  imagesset  36344