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Theorem funimage 32624
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 3960 . . . 4 ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝐴) ⊗ V))) ⊆ (V × V)
2 df-rel 5362 . . . 4 (Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝐴) ⊗ V))) ↔ ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝐴) ⊗ V))) ⊆ (V × V))
31, 2mpbir 223 . . 3 Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝐴) ⊗ V)))
4 df-image 32560 . . . 4 Image𝐴 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝐴) ⊗ V)))
54releqi 5450 . . 3 (Rel Image𝐴 ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝐴) ⊗ V))))
63, 5mpbir 223 . 2 Rel Image𝐴
7 vex 3401 . . . . . 6 𝑥 ∈ V
8 vex 3401 . . . . . 6 𝑦 ∈ V
97, 8brimage 32622 . . . . 5 (𝑥Image𝐴𝑦𝑦 = (𝐴𝑥))
10 vex 3401 . . . . . 6 𝑧 ∈ V
117, 10brimage 32622 . . . . 5 (𝑥Image𝐴𝑧𝑧 = (𝐴𝑥))
12 eqtr3 2801 . . . . 5 ((𝑦 = (𝐴𝑥) ∧ 𝑧 = (𝐴𝑥)) → 𝑦 = 𝑧)
139, 11, 12syl2anb 591 . . . 4 ((𝑥Image𝐴𝑦𝑥Image𝐴𝑧) → 𝑦 = 𝑧)
1413gen2 1840 . . 3 𝑦𝑧((𝑥Image𝐴𝑦𝑥Image𝐴𝑧) → 𝑦 = 𝑧)
1514ax-gen 1839 . 2 𝑥𝑦𝑧((𝑥Image𝐴𝑦𝑥Image𝐴𝑧) → 𝑦 = 𝑧)
16 dffun2 6145 . 2 (Fun Image𝐴 ↔ (Rel Image𝐴 ∧ ∀𝑥𝑦𝑧((𝑥Image𝐴𝑦𝑥Image𝐴𝑧) → 𝑦 = 𝑧)))
176, 15, 16mpbir2an 701 1 Fun Image𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wal 1599   = wceq 1601  Vcvv 3398  cdif 3789  wss 3792  csymdif 4066   class class class wbr 4886   E cep 5265   × cxp 5353  ccnv 5354  ran crn 5356  cima 5358  ccom 5359  Rel wrel 5360  Fun wfun 6129  ctxp 32526  Imagecimage 32536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-symdif 4067  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-eprel 5266  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fo 6141  df-fv 6143  df-1st 7445  df-2nd 7446  df-txp 32550  df-image 32560
This theorem is referenced by:  fnimage  32625  imageval  32626  imagesset  32649
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