Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfimafnf Structured version   Visualization version   GIF version

Theorem dfimafnf 32653
Description: Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Thierry Arnoux, 24-Apr-2017.)
Hypotheses
Ref Expression
dfimafnf.1 𝑥𝐴
dfimafnf.2 𝑥𝐹
Assertion
Ref Expression
dfimafnf ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑦,𝐹
Allowed substitution hints:   𝐴(𝑥)   𝐹(𝑥)

Proof of Theorem dfimafnf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dfima2 6082 . . 3 (𝐹𝐴) = {𝑦 ∣ ∃𝑧𝐴 𝑧𝐹𝑦}
2 ssel 3989 . . . . . . 7 (𝐴 ⊆ dom 𝐹 → (𝑧𝐴𝑧 ∈ dom 𝐹))
3 eqcom 2742 . . . . . . . . 9 ((𝐹𝑧) = 𝑦𝑦 = (𝐹𝑧))
4 funbrfvb 6962 . . . . . . . . 9 ((Fun 𝐹𝑧 ∈ dom 𝐹) → ((𝐹𝑧) = 𝑦𝑧𝐹𝑦))
53, 4bitr3id 285 . . . . . . . 8 ((Fun 𝐹𝑧 ∈ dom 𝐹) → (𝑦 = (𝐹𝑧) ↔ 𝑧𝐹𝑦))
65ex 412 . . . . . . 7 (Fun 𝐹 → (𝑧 ∈ dom 𝐹 → (𝑦 = (𝐹𝑧) ↔ 𝑧𝐹𝑦)))
72, 6syl9r 78 . . . . . 6 (Fun 𝐹 → (𝐴 ⊆ dom 𝐹 → (𝑧𝐴 → (𝑦 = (𝐹𝑧) ↔ 𝑧𝐹𝑦))))
87imp31 417 . . . . 5 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ 𝑧𝐴) → (𝑦 = (𝐹𝑧) ↔ 𝑧𝐹𝑦))
98rexbidva 3175 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (∃𝑧𝐴 𝑦 = (𝐹𝑧) ↔ ∃𝑧𝐴 𝑧𝐹𝑦))
109abbidv 2806 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → {𝑦 ∣ ∃𝑧𝐴 𝑦 = (𝐹𝑧)} = {𝑦 ∣ ∃𝑧𝐴 𝑧𝐹𝑦})
111, 10eqtr4id 2794 . 2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑧𝐴 𝑦 = (𝐹𝑧)})
12 nfcv 2903 . . . 4 𝑧𝐴
13 dfimafnf.1 . . . 4 𝑥𝐴
14 dfimafnf.2 . . . . . 6 𝑥𝐹
15 nfcv 2903 . . . . . 6 𝑥𝑧
1614, 15nffv 6917 . . . . 5 𝑥(𝐹𝑧)
1716nfeq2 2921 . . . 4 𝑥 𝑦 = (𝐹𝑧)
18 nfv 1912 . . . 4 𝑧 𝑦 = (𝐹𝑥)
19 fveq2 6907 . . . . 5 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
2019eqeq2d 2746 . . . 4 (𝑧 = 𝑥 → (𝑦 = (𝐹𝑧) ↔ 𝑦 = (𝐹𝑥)))
2112, 13, 17, 18, 20cbvrexfw 3303 . . 3 (∃𝑧𝐴 𝑦 = (𝐹𝑧) ↔ ∃𝑥𝐴 𝑦 = (𝐹𝑥))
2221abbii 2807 . 2 {𝑦 ∣ ∃𝑧𝐴 𝑦 = (𝐹𝑧)} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
2311, 22eqtrdi 2791 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {cab 2712  wnfc 2888  wrex 3068  wss 3963   class class class wbr 5148  dom cdm 5689  cima 5692  Fun wfun 6557  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571
This theorem is referenced by:  funimass4f  32654
  Copyright terms: Public domain W3C validator