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Theorem dfimafnf 32646
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 6080 . . 3 (𝐹𝐴) = {𝑦 ∣ ∃𝑧𝐴 𝑧𝐹𝑦}
2 ssel 3977 . . . . . . 7 (𝐴 ⊆ dom 𝐹 → (𝑧𝐴𝑧 ∈ dom 𝐹))
3 eqcom 2744 . . . . . . . . 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 3177 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (∃𝑧𝐴 𝑦 = (𝐹𝑧) ↔ ∃𝑧𝐴 𝑧𝐹𝑦))
109abbidv 2808 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → {𝑦 ∣ ∃𝑧𝐴 𝑦 = (𝐹𝑧)} = {𝑦 ∣ ∃𝑧𝐴 𝑧𝐹𝑦})
111, 10eqtr4id 2796 . 2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑧𝐴 𝑦 = (𝐹𝑧)})
12 nfcv 2905 . . . 4 𝑧𝐴
13 dfimafnf.1 . . . 4 𝑥𝐴
14 dfimafnf.2 . . . . . 6 𝑥𝐹
15 nfcv 2905 . . . . . 6 𝑥𝑧
1614, 15nffv 6916 . . . . 5 𝑥(𝐹𝑧)
1716nfeq2 2923 . . . 4 𝑥 𝑦 = (𝐹𝑧)
18 nfv 1914 . . . 4 𝑧 𝑦 = (𝐹𝑥)
19 fveq2 6906 . . . . 5 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
2019eqeq2d 2748 . . . 4 (𝑧 = 𝑥 → (𝑦 = (𝐹𝑧) ↔ 𝑦 = (𝐹𝑥)))
2112, 13, 17, 18, 20cbvrexfw 3305 . . 3 (∃𝑧𝐴 𝑦 = (𝐹𝑧) ↔ ∃𝑥𝐴 𝑦 = (𝐹𝑥))
2221abbii 2809 . 2 {𝑦 ∣ ∃𝑧𝐴 𝑦 = (𝐹𝑧)} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
2311, 22eqtrdi 2793 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 1540  wcel 2108  {cab 2714  wnfc 2890  wrex 3070  wss 3951   class class class wbr 5143  dom cdm 5685  cima 5688  Fun wfun 6555  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  funimass4f  32647
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