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Theorem inisegn0 6115
Description: Nonemptiness of an initial segment in terms of range. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Assertion
Ref Expression
inisegn0 (𝐴 ∈ ran 𝐹 ↔ (𝐹 “ {𝐴}) ≠ ∅)

Proof of Theorem inisegn0
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3500 . 2 (𝐴 ∈ ran 𝐹𝐴 ∈ V)
2 snprc 4716 . . . . . 6 𝐴 ∈ V ↔ {𝐴} = ∅)
32biimpi 216 . . . . 5 𝐴 ∈ V → {𝐴} = ∅)
43imaeq2d 6077 . . . 4 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅))
5 ima0 6094 . . . 4 (𝐹 “ ∅) = ∅
64, 5eqtrdi 2792 . . 3 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅)
76necon1ai 2967 . 2 ((𝐹 “ {𝐴}) ≠ ∅ → 𝐴 ∈ V)
8 eleq1 2828 . . 3 (𝑎 = 𝐴 → (𝑎 ∈ ran 𝐹𝐴 ∈ ran 𝐹))
9 sneq 4635 . . . . 5 (𝑎 = 𝐴 → {𝑎} = {𝐴})
109imaeq2d 6077 . . . 4 (𝑎 = 𝐴 → (𝐹 “ {𝑎}) = (𝐹 “ {𝐴}))
1110neeq1d 2999 . . 3 (𝑎 = 𝐴 → ((𝐹 “ {𝑎}) ≠ ∅ ↔ (𝐹 “ {𝐴}) ≠ ∅))
12 abn0 4384 . . . 4 ({𝑏𝑏𝐹𝑎} ≠ ∅ ↔ ∃𝑏 𝑏𝐹𝑎)
13 iniseg 6114 . . . . . 6 (𝑎 ∈ V → (𝐹 “ {𝑎}) = {𝑏𝑏𝐹𝑎})
1413elv 3484 . . . . 5 (𝐹 “ {𝑎}) = {𝑏𝑏𝐹𝑎}
1514neeq1i 3004 . . . 4 ((𝐹 “ {𝑎}) ≠ ∅ ↔ {𝑏𝑏𝐹𝑎} ≠ ∅)
16 vex 3483 . . . . 5 𝑎 ∈ V
1716elrn 5903 . . . 4 (𝑎 ∈ ran 𝐹 ↔ ∃𝑏 𝑏𝐹𝑎)
1812, 15, 173bitr4ri 304 . . 3 (𝑎 ∈ ran 𝐹 ↔ (𝐹 “ {𝑎}) ≠ ∅)
198, 11, 18vtoclbg 3556 . 2 (𝐴 ∈ V → (𝐴 ∈ ran 𝐹 ↔ (𝐹 “ {𝐴}) ≠ ∅))
201, 7, 19pm5.21nii 378 1 (𝐴 ∈ ran 𝐹 ↔ (𝐹 “ {𝐴}) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206   = wceq 1539  wex 1778  wcel 2107  {cab 2713  wne 2939  Vcvv 3479  c0 4332  {csn 4625   class class class wbr 5142  ccnv 5683  ran crn 5685  cima 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697
This theorem is referenced by:  fnpreimac  32682  dnnumch3lem  43063  dnnumch3  43064  wessf1ornlem  45195
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