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Theorem dfimafnf 32502
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 6066 . . 3 (𝐹𝐴) = {𝑦 ∣ ∃𝑧𝐴 𝑧𝐹𝑦}
2 ssel 3970 . . . . . . 7 (𝐴 ⊆ dom 𝐹 → (𝑧𝐴𝑧 ∈ dom 𝐹))
3 eqcom 2732 . . . . . . . . 9 ((𝐹𝑧) = 𝑦𝑦 = (𝐹𝑧))
4 funbrfvb 6951 . . . . . . . . 9 ((Fun 𝐹𝑧 ∈ dom 𝐹) → ((𝐹𝑧) = 𝑦𝑧𝐹𝑦))
53, 4bitr3id 284 . . . . . . . 8 ((Fun 𝐹𝑧 ∈ dom 𝐹) → (𝑦 = (𝐹𝑧) ↔ 𝑧𝐹𝑦))
65ex 411 . . . . . . 7 (Fun 𝐹 → (𝑧 ∈ dom 𝐹 → (𝑦 = (𝐹𝑧) ↔ 𝑧𝐹𝑦)))
72, 6syl9r 78 . . . . . 6 (Fun 𝐹 → (𝐴 ⊆ dom 𝐹 → (𝑧𝐴 → (𝑦 = (𝐹𝑧) ↔ 𝑧𝐹𝑦))))
87imp31 416 . . . . 5 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ 𝑧𝐴) → (𝑦 = (𝐹𝑧) ↔ 𝑧𝐹𝑦))
98rexbidva 3166 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (∃𝑧𝐴 𝑦 = (𝐹𝑧) ↔ ∃𝑧𝐴 𝑧𝐹𝑦))
109abbidv 2794 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → {𝑦 ∣ ∃𝑧𝐴 𝑦 = (𝐹𝑧)} = {𝑦 ∣ ∃𝑧𝐴 𝑧𝐹𝑦})
111, 10eqtr4id 2784 . 2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑧𝐴 𝑦 = (𝐹𝑧)})
12 nfcv 2891 . . . 4 𝑧𝐴
13 dfimafnf.1 . . . 4 𝑥𝐴
14 dfimafnf.2 . . . . . 6 𝑥𝐹
15 nfcv 2891 . . . . . 6 𝑥𝑧
1614, 15nffv 6906 . . . . 5 𝑥(𝐹𝑧)
1716nfeq2 2909 . . . 4 𝑥 𝑦 = (𝐹𝑧)
18 nfv 1909 . . . 4 𝑧 𝑦 = (𝐹𝑥)
19 fveq2 6896 . . . . 5 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
2019eqeq2d 2736 . . . 4 (𝑧 = 𝑥 → (𝑦 = (𝐹𝑧) ↔ 𝑦 = (𝐹𝑥)))
2112, 13, 17, 18, 20cbvrexfw 3292 . . 3 (∃𝑧𝐴 𝑦 = (𝐹𝑧) ↔ ∃𝑥𝐴 𝑦 = (𝐹𝑥))
2221abbii 2795 . 2 {𝑦 ∣ ∃𝑧𝐴 𝑦 = (𝐹𝑧)} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
2311, 22eqtrdi 2781 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  {cab 2702  wnfc 2875  wrex 3059  wss 3944   class class class wbr 5149  dom cdm 5678  cima 5681  Fun wfun 6543  cfv 6549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-fv 6557
This theorem is referenced by:  funimass4f  32503
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