Theorem brimage 36140

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 36078	. 2 ⊢ Image𝑅 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ ◡𝑅) ⊗ V)))
4		brxp 5681	. . 3 ⊢ (𝐴(V × V)𝐵 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
5	1, 2, 4	mpbir2an 712	. 2 ⊢ 𝐴(V × V)𝐵
6		vex 3446	. . . . 5 ⊢ 𝑥 ∈ V
7		vex 3446	. . . . 5 ⊢ 𝑦 ∈ V
8	6, 7	brcnv 5839	. . . 4 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
9	8	rexbii 3085	. . 3 ⊢ (∃𝑦 ∈ 𝐴 𝑥◡𝑅𝑦 ↔ ∃𝑦 ∈ 𝐴 𝑦𝑅𝑥)
10	6, 1	coep 35968	. . 3 ⊢ (𝑥( E ∘ ◡𝑅)𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥◡𝑅𝑦)
11	6	elima 6032	. . 3 ⊢ (𝑥 ∈ (𝑅 “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑦𝑅𝑥)
12	9, 10, 11	3bitr4ri 304	. 2 ⊢ (𝑥 ∈ (𝑅 “ 𝐴) ↔ 𝑥( E ∘ ◡𝑅)𝐴)
13	1, 2, 3, 5, 12	brtxpsd3 36110	1 ⊢ (𝐴Image𝑅𝐵 ↔ 𝐵 = (𝑅 “ 𝐴))

Syntax hints:   ↔ wb 206   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  Vcvv 3442   class class class wbr 5100   E cep 5531   × cxp 5630  ^◡ccnv 5631   “ cima 5635   ∘ ccom 5636  Imagecimage 36054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-symdif 4207  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-eprel 5532  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-1st 7943  df-2nd 7944  df-txp 36068  df-image 36078

This theorem is referenced by:  brimageg  36141  funimage  36142  fnimage  36143  imageval  36144  brdomain  36147  brrange  36148  brimg  36151  funpartlem  36158  imagesset  36169