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Theorem inisegn0 6097
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 3492 . 2 (𝐴 ∈ ran 𝐹𝐴 ∈ V)
2 snprc 4721 . . . . . 6 𝐴 ∈ V ↔ {𝐴} = ∅)
32biimpi 215 . . . . 5 𝐴 ∈ V → {𝐴} = ∅)
43imaeq2d 6059 . . . 4 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅))
5 ima0 6076 . . . 4 (𝐹 “ ∅) = ∅
64, 5eqtrdi 2788 . . 3 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅)
76necon1ai 2968 . 2 ((𝐹 “ {𝐴}) ≠ ∅ → 𝐴 ∈ V)
8 eleq1 2821 . . 3 (𝑎 = 𝐴 → (𝑎 ∈ ran 𝐹𝐴 ∈ ran 𝐹))
9 sneq 4638 . . . . 5 (𝑎 = 𝐴 → {𝑎} = {𝐴})
109imaeq2d 6059 . . . 4 (𝑎 = 𝐴 → (𝐹 “ {𝑎}) = (𝐹 “ {𝐴}))
1110neeq1d 3000 . . 3 (𝑎 = 𝐴 → ((𝐹 “ {𝑎}) ≠ ∅ ↔ (𝐹 “ {𝐴}) ≠ ∅))
12 abn0 4380 . . . 4 ({𝑏𝑏𝐹𝑎} ≠ ∅ ↔ ∃𝑏 𝑏𝐹𝑎)
13 iniseg 6096 . . . . . 6 (𝑎 ∈ V → (𝐹 “ {𝑎}) = {𝑏𝑏𝐹𝑎})
1413elv 3480 . . . . 5 (𝐹 “ {𝑎}) = {𝑏𝑏𝐹𝑎}
1514neeq1i 3005 . . . 4 ((𝐹 “ {𝑎}) ≠ ∅ ↔ {𝑏𝑏𝐹𝑎} ≠ ∅)
16 vex 3478 . . . . 5 𝑎 ∈ V
1716elrn 5893 . . . 4 (𝑎 ∈ ran 𝐹 ↔ ∃𝑏 𝑏𝐹𝑎)
1812, 15, 173bitr4ri 303 . . 3 (𝑎 ∈ ran 𝐹 ↔ (𝐹 “ {𝑎}) ≠ ∅)
198, 11, 18vtoclbg 3559 . 2 (𝐴 ∈ V → (𝐴 ∈ ran 𝐹 ↔ (𝐹 “ {𝐴}) ≠ ∅))
201, 7, 19pm5.21nii 379 1 (𝐴 ∈ ran 𝐹 ↔ (𝐹 “ {𝐴}) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1541  wex 1781  wcel 2106  {cab 2709  wne 2940  Vcvv 3474  c0 4322  {csn 4628   class class class wbr 5148  ccnv 5675  ran crn 5677  cima 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689
This theorem is referenced by:  fnpreimac  31891  dnnumch3lem  41778  dnnumch3  41779  wessf1ornlem  43872
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