Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  funimage Structured version   Visualization version   GIF version

Theorem funimage 32019
Description: Image𝐴 is a function. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
funimage Fun Image𝐴

Proof of Theorem funimage
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 difss 3735 . . . 4 ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝐴) ⊗ V))) ⊆ (V × V)
2 df-rel 5119 . . . 4 (Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝐴) ⊗ V))) ↔ ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝐴) ⊗ V))) ⊆ (V × V))
31, 2mpbir 221 . . 3 Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝐴) ⊗ V)))
4 df-image 31955 . . . 4 Image𝐴 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝐴) ⊗ V)))
54releqi 5200 . . 3 (Rel Image𝐴 ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝐴) ⊗ V))))
63, 5mpbir 221 . 2 Rel Image𝐴
7 vex 3201 . . . . . 6 𝑥 ∈ V
8 vex 3201 . . . . . 6 𝑦 ∈ V
97, 8brimage 32017 . . . . 5 (𝑥Image𝐴𝑦𝑦 = (𝐴𝑥))
10 vex 3201 . . . . . 6 𝑧 ∈ V
117, 10brimage 32017 . . . . 5 (𝑥Image𝐴𝑧𝑧 = (𝐴𝑥))
12 eqtr3 2642 . . . . 5 ((𝑦 = (𝐴𝑥) ∧ 𝑧 = (𝐴𝑥)) → 𝑦 = 𝑧)
139, 11, 12syl2anb 496 . . . 4 ((𝑥Image𝐴𝑦𝑥Image𝐴𝑧) → 𝑦 = 𝑧)
1413gen2 1722 . . 3 𝑦𝑧((𝑥Image𝐴𝑦𝑥Image𝐴𝑧) → 𝑦 = 𝑧)
1514ax-gen 1721 . 2 𝑥𝑦𝑧((𝑥Image𝐴𝑦𝑥Image𝐴𝑧) → 𝑦 = 𝑧)
16 dffun2 5896 . 2 (Fun Image𝐴 ↔ (Rel Image𝐴 ∧ ∀𝑥𝑦𝑧((𝑥Image𝐴𝑦𝑥Image𝐴𝑧) → 𝑦 = 𝑧)))
176, 15, 16mpbir2an 955 1 Fun Image𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1480   = wceq 1482  Vcvv 3198  cdif 3569  wss 3572  csymdif 3841   class class class wbr 4651   E cep 5026   × cxp 5110  ccnv 5111  ran crn 5113  cima 5115  ccom 5116  Rel wrel 5117  Fun wfun 5880  ctxp 31921  Imagecimage 31931
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-symdif 3842  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-eprel 5027  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fo 5892  df-fv 5894  df-1st 7165  df-2nd 7166  df-txp 31945  df-image 31955
This theorem is referenced by:  fnimage  32020  imageval  32021  imagesset  32044
  Copyright terms: Public domain W3C validator