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Theorem inisegn0 5402
Description: Nonemptyness 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 3184 . 2 (𝐴 ∈ ran 𝐹𝐴 ∈ V)
2 snprc 4196 . . . . . 6 𝐴 ∈ V ↔ {𝐴} = ∅)
32biimpi 204 . . . . 5 𝐴 ∈ V → {𝐴} = ∅)
43imaeq2d 5371 . . . 4 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅))
5 ima0 5386 . . . 4 (𝐹 “ ∅) = ∅
64, 5syl6eq 2659 . . 3 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅)
76necon1ai 2808 . 2 ((𝐹 “ {𝐴}) ≠ ∅ → 𝐴 ∈ V)
8 eleq1 2675 . . 3 (𝑎 = 𝐴 → (𝑎 ∈ ran 𝐹𝐴 ∈ ran 𝐹))
9 sneq 4134 . . . . 5 (𝑎 = 𝐴 → {𝑎} = {𝐴})
109imaeq2d 5371 . . . 4 (𝑎 = 𝐴 → (𝐹 “ {𝑎}) = (𝐹 “ {𝐴}))
1110neeq1d 2840 . . 3 (𝑎 = 𝐴 → ((𝐹 “ {𝑎}) ≠ ∅ ↔ (𝐹 “ {𝐴}) ≠ ∅))
12 abn0 3907 . . . 4 ({𝑏𝑏𝐹𝑎} ≠ ∅ ↔ ∃𝑏 𝑏𝐹𝑎)
13 vex 3175 . . . . . 6 𝑎 ∈ V
14 iniseg 5401 . . . . . 6 (𝑎 ∈ V → (𝐹 “ {𝑎}) = {𝑏𝑏𝐹𝑎})
1513, 14ax-mp 5 . . . . 5 (𝐹 “ {𝑎}) = {𝑏𝑏𝐹𝑎}
1615neeq1i 2845 . . . 4 ((𝐹 “ {𝑎}) ≠ ∅ ↔ {𝑏𝑏𝐹𝑎} ≠ ∅)
1713elrn 5273 . . . 4 (𝑎 ∈ ran 𝐹 ↔ ∃𝑏 𝑏𝐹𝑎)
1812, 16, 173bitr4ri 291 . . 3 (𝑎 ∈ ran 𝐹 ↔ (𝐹 “ {𝑎}) ≠ ∅)
198, 11, 18vtoclbg 3239 . 2 (𝐴 ∈ V → (𝐴 ∈ ran 𝐹 ↔ (𝐹 “ {𝐴}) ≠ ∅))
201, 7, 19pm5.21nii 366 1 (𝐴 ∈ ran 𝐹 ↔ (𝐹 “ {𝐴}) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 194   = wceq 1474  wex 1694  wcel 1976  {cab 2595  wne 2779  Vcvv 3172  c0 3873  {csn 4124   class class class wbr 4577  ccnv 5026  ran crn 5028  cima 5030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-xp 5033  df-cnv 5035  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040
This theorem is referenced by:  dnnumch3lem  36417  dnnumch3  36418  wessf1ornlem  38149
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