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Theorem funimage 33286
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 4105 . . . 4 ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝐴) ⊗ V))) ⊆ (V × V)
2 df-rel 5555 . . . 4 (Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝐴) ⊗ V))) ↔ ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝐴) ⊗ V))) ⊆ (V × V))
31, 2mpbir 232 . . 3 Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝐴) ⊗ V)))
4 df-image 33222 . . . 4 Image𝐴 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝐴) ⊗ V)))
54releqi 5645 . . 3 (Rel Image𝐴 ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝐴) ⊗ V))))
63, 5mpbir 232 . 2 Rel Image𝐴
7 vex 3495 . . . . . 6 𝑥 ∈ V
8 vex 3495 . . . . . 6 𝑦 ∈ V
97, 8brimage 33284 . . . . 5 (𝑥Image𝐴𝑦𝑦 = (𝐴𝑥))
10 vex 3495 . . . . . 6 𝑧 ∈ V
117, 10brimage 33284 . . . . 5 (𝑥Image𝐴𝑧𝑧 = (𝐴𝑥))
12 eqtr3 2840 . . . . 5 ((𝑦 = (𝐴𝑥) ∧ 𝑧 = (𝐴𝑥)) → 𝑦 = 𝑧)
139, 11, 12syl2anb 597 . . . 4 ((𝑥Image𝐴𝑦𝑥Image𝐴𝑧) → 𝑦 = 𝑧)
1413gen2 1788 . . 3 𝑦𝑧((𝑥Image𝐴𝑦𝑥Image𝐴𝑧) → 𝑦 = 𝑧)
1514ax-gen 1787 . 2 𝑥𝑦𝑧((𝑥Image𝐴𝑦𝑥Image𝐴𝑧) → 𝑦 = 𝑧)
16 dffun2 6358 . 2 (Fun Image𝐴 ↔ (Rel Image𝐴 ∧ ∀𝑥𝑦𝑧((𝑥Image𝐴𝑦𝑥Image𝐴𝑧) → 𝑦 = 𝑧)))
176, 15, 16mpbir2an 707 1 Fun Image𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1526   = wceq 1528  Vcvv 3492  cdif 3930  wss 3933  csymdif 4215   class class class wbr 5057   E cep 5457   × cxp 5546  ccnv 5547  ran crn 5549  cima 5551  ccom 5552  Rel wrel 5553  Fun wfun 6342  ctxp 33188  Imagecimage 33198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-symdif 4216  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-eprel 5458  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-1st 7678  df-2nd 7679  df-txp 33212  df-image 33222
This theorem is referenced by:  fnimage  33287  imageval  33288  imagesset  33311
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