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Theorem funimage 32372
 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 3888 . . . 4 ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝐴) ⊗ V))) ⊆ (V × V)
2 df-rel 5256 . . . 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 32308 . . . 4 Image𝐴 = ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝐴) ⊗ V)))
54releqi 5342 . . 3 (Rel Image𝐴 ↔ Rel ((V × V) ∖ ran ((V ⊗ E ) △ (( E ∘ 𝐴) ⊗ V))))
63, 5mpbir 221 . 2 Rel Image𝐴
7 vex 3354 . . . . . 6 𝑥 ∈ V
8 vex 3354 . . . . . 6 𝑦 ∈ V
97, 8brimage 32370 . . . . 5 (𝑥Image𝐴𝑦𝑦 = (𝐴𝑥))
10 vex 3354 . . . . . 6 𝑧 ∈ V
117, 10brimage 32370 . . . . 5 (𝑥Image𝐴𝑧𝑧 = (𝐴𝑥))
12 eqtr3 2792 . . . . 5 ((𝑦 = (𝐴𝑥) ∧ 𝑧 = (𝐴𝑥)) → 𝑦 = 𝑧)
139, 11, 12syl2anb 585 . . . 4 ((𝑥Image𝐴𝑦𝑥Image𝐴𝑧) → 𝑦 = 𝑧)
1413gen2 1871 . . 3 𝑦𝑧((𝑥Image𝐴𝑦𝑥Image𝐴𝑧) → 𝑦 = 𝑧)
1514ax-gen 1870 . 2 𝑥𝑦𝑧((𝑥Image𝐴𝑦𝑥Image𝐴𝑧) → 𝑦 = 𝑧)
16 dffun2 6041 . 2 (Fun Image𝐴 ↔ (Rel Image𝐴 ∧ ∀𝑥𝑦𝑧((𝑥Image𝐴𝑦𝑥Image𝐴𝑧) → 𝑦 = 𝑧)))
176, 15, 16mpbir2an 690 1 Fun Image𝐴
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382  ∀wal 1629   = wceq 1631  Vcvv 3351   ∖ cdif 3720   ⊆ wss 3723   △ csymdif 3992   class class class wbr 4786   E cep 5161   × cxp 5247  ◡ccnv 5248  ran crn 5250   “ cima 5252   ∘ ccom 5253  Rel wrel 5254  Fun wfun 6025   ⊗ ctxp 32274  Imagecimage 32284 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-symdif 3993  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-eprel 5162  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-1st 7315  df-2nd 7316  df-txp 32298  df-image 32308 This theorem is referenced by:  fnimage  32373  imageval  32374  imagesset  32397
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