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

Theorem dfimafnf 29409
 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 ssel 3589 . . . . . . 7 (𝐴 ⊆ dom 𝐹 → (𝑧𝐴𝑧 ∈ dom 𝐹))
2 eqcom 2627 . . . . . . . . 9 ((𝐹𝑧) = 𝑦𝑦 = (𝐹𝑧))
3 funbrfvb 6225 . . . . . . . . 9 ((Fun 𝐹𝑧 ∈ dom 𝐹) → ((𝐹𝑧) = 𝑦𝑧𝐹𝑦))
42, 3syl5bbr 274 . . . . . . . 8 ((Fun 𝐹𝑧 ∈ dom 𝐹) → (𝑦 = (𝐹𝑧) ↔ 𝑧𝐹𝑦))
54ex 450 . . . . . . 7 (Fun 𝐹 → (𝑧 ∈ dom 𝐹 → (𝑦 = (𝐹𝑧) ↔ 𝑧𝐹𝑦)))
61, 5syl9r 78 . . . . . 6 (Fun 𝐹 → (𝐴 ⊆ dom 𝐹 → (𝑧𝐴 → (𝑦 = (𝐹𝑧) ↔ 𝑧𝐹𝑦))))
76imp31 448 . . . . 5 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ 𝑧𝐴) → (𝑦 = (𝐹𝑧) ↔ 𝑧𝐹𝑦))
87rexbidva 3045 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (∃𝑧𝐴 𝑦 = (𝐹𝑧) ↔ ∃𝑧𝐴 𝑧𝐹𝑦))
98abbidv 2739 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → {𝑦 ∣ ∃𝑧𝐴 𝑦 = (𝐹𝑧)} = {𝑦 ∣ ∃𝑧𝐴 𝑧𝐹𝑦})
10 dfima2 5456 . . 3 (𝐹𝐴) = {𝑦 ∣ ∃𝑧𝐴 𝑧𝐹𝑦}
119, 10syl6reqr 2673 . 2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑧𝐴 𝑦 = (𝐹𝑧)})
12 nfcv 2762 . . . 4 𝑧𝐴
13 dfimafnf.1 . . . 4 𝑥𝐴
14 dfimafnf.2 . . . . . 6 𝑥𝐹
15 nfcv 2762 . . . . . 6 𝑥𝑧
1614, 15nffv 6185 . . . . 5 𝑥(𝐹𝑧)
1716nfeq2 2777 . . . 4 𝑥 𝑦 = (𝐹𝑧)
18 nfv 1841 . . . 4 𝑧 𝑦 = (𝐹𝑥)
19 fveq2 6178 . . . . 5 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
2019eqeq2d 2630 . . . 4 (𝑧 = 𝑥 → (𝑦 = (𝐹𝑧) ↔ 𝑦 = (𝐹𝑥)))
2112, 13, 17, 18, 20cbvrexf 3161 . . 3 (∃𝑧𝐴 𝑦 = (𝐹𝑧) ↔ ∃𝑥𝐴 𝑦 = (𝐹𝑥))
2221abbii 2737 . 2 {𝑦 ∣ ∃𝑧𝐴 𝑦 = (𝐹𝑧)} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
2311, 22syl6eq 2670 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1481   ∈ wcel 1988  {cab 2606  Ⅎwnfc 2749  ∃wrex 2910   ⊆ wss 3567   class class class wbr 4644  dom cdm 5104   “ cima 5107  Fun wfun 5870  ‘cfv 5876 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-fv 5884 This theorem is referenced by:  funimass4f  29410
 Copyright terms: Public domain W3C validator