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Theorem dfimafnf 30384
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 3964 . . . . . . 7 (𝐴 ⊆ dom 𝐹 → (𝑧𝐴𝑧 ∈ dom 𝐹))
2 eqcom 2831 . . . . . . . . 9 ((𝐹𝑧) = 𝑦𝑦 = (𝐹𝑧))
3 funbrfvb 6723 . . . . . . . . 9 ((Fun 𝐹𝑧 ∈ dom 𝐹) → ((𝐹𝑧) = 𝑦𝑧𝐹𝑦))
42, 3syl5bbr 287 . . . . . . . 8 ((Fun 𝐹𝑧 ∈ dom 𝐹) → (𝑦 = (𝐹𝑧) ↔ 𝑧𝐹𝑦))
54ex 415 . . . . . . 7 (Fun 𝐹 → (𝑧 ∈ dom 𝐹 → (𝑦 = (𝐹𝑧) ↔ 𝑧𝐹𝑦)))
61, 5syl9r 78 . . . . . 6 (Fun 𝐹 → (𝐴 ⊆ dom 𝐹 → (𝑧𝐴 → (𝑦 = (𝐹𝑧) ↔ 𝑧𝐹𝑦))))
76imp31 420 . . . . 5 (((Fun 𝐹𝐴 ⊆ dom 𝐹) ∧ 𝑧𝐴) → (𝑦 = (𝐹𝑧) ↔ 𝑧𝐹𝑦))
87rexbidva 3299 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (∃𝑧𝐴 𝑦 = (𝐹𝑧) ↔ ∃𝑧𝐴 𝑧𝐹𝑦))
98abbidv 2888 . . 3 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → {𝑦 ∣ ∃𝑧𝐴 𝑦 = (𝐹𝑧)} = {𝑦 ∣ ∃𝑧𝐴 𝑧𝐹𝑦})
10 dfima2 5934 . . 3 (𝐹𝐴) = {𝑦 ∣ ∃𝑧𝐴 𝑧𝐹𝑦}
119, 10syl6reqr 2878 . 2 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑧𝐴 𝑦 = (𝐹𝑧)})
12 nfcv 2980 . . . 4 𝑧𝐴
13 dfimafnf.1 . . . 4 𝑥𝐴
14 dfimafnf.2 . . . . . 6 𝑥𝐹
15 nfcv 2980 . . . . . 6 𝑥𝑧
1614, 15nffv 6683 . . . . 5 𝑥(𝐹𝑧)
1716nfeq2 2998 . . . 4 𝑥 𝑦 = (𝐹𝑧)
18 nfv 1914 . . . 4 𝑧 𝑦 = (𝐹𝑥)
19 fveq2 6673 . . . . 5 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
2019eqeq2d 2835 . . . 4 (𝑧 = 𝑥 → (𝑦 = (𝐹𝑧) ↔ 𝑦 = (𝐹𝑥)))
2112, 13, 17, 18, 20cbvrexfw 3441 . . 3 (∃𝑧𝐴 𝑦 = (𝐹𝑧) ↔ ∃𝑥𝐴 𝑦 = (𝐹𝑥))
2221abbii 2889 . 2 {𝑦 ∣ ∃𝑧𝐴 𝑦 = (𝐹𝑧)} = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)}
2311, 22syl6eq 2875 1 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1536  wcel 2113  {cab 2802  wnfc 2964  wrex 3142  wss 3939   class class class wbr 5069  dom cdm 5558  cima 5561  Fun wfun 6352  cfv 6358
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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-fv 6366
This theorem is referenced by:  funimass4f  30385
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