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Theorem inisegn0 5937
 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 3428 . 2 (𝐴 ∈ ran 𝐹𝐴 ∈ V)
2 snprc 4613 . . . . . 6 𝐴 ∈ V ↔ {𝐴} = ∅)
32biimpi 219 . . . . 5 𝐴 ∈ V → {𝐴} = ∅)
43imaeq2d 5905 . . . 4 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅))
5 ima0 5921 . . . 4 (𝐹 “ ∅) = ∅
64, 5eqtrdi 2809 . . 3 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅)
76necon1ai 2978 . 2 ((𝐹 “ {𝐴}) ≠ ∅ → 𝐴 ∈ V)
8 eleq1 2839 . . 3 (𝑎 = 𝐴 → (𝑎 ∈ ran 𝐹𝐴 ∈ ran 𝐹))
9 sneq 4535 . . . . 5 (𝑎 = 𝐴 → {𝑎} = {𝐴})
109imaeq2d 5905 . . . 4 (𝑎 = 𝐴 → (𝐹 “ {𝑎}) = (𝐹 “ {𝐴}))
1110neeq1d 3010 . . 3 (𝑎 = 𝐴 → ((𝐹 “ {𝑎}) ≠ ∅ ↔ (𝐹 “ {𝐴}) ≠ ∅))
12 abn0 4280 . . . 4 ({𝑏𝑏𝐹𝑎} ≠ ∅ ↔ ∃𝑏 𝑏𝐹𝑎)
13 iniseg 5936 . . . . . 6 (𝑎 ∈ V → (𝐹 “ {𝑎}) = {𝑏𝑏𝐹𝑎})
1413elv 3415 . . . . 5 (𝐹 “ {𝑎}) = {𝑏𝑏𝐹𝑎}
1514neeq1i 3015 . . . 4 ((𝐹 “ {𝑎}) ≠ ∅ ↔ {𝑏𝑏𝐹𝑎} ≠ ∅)
16 vex 3413 . . . . 5 𝑎 ∈ V
1716elrn 5738 . . . 4 (𝑎 ∈ ran 𝐹 ↔ ∃𝑏 𝑏𝐹𝑎)
1812, 15, 173bitr4ri 307 . . 3 (𝑎 ∈ ran 𝐹 ↔ (𝐹 “ {𝑎}) ≠ ∅)
198, 11, 18vtoclbg 3489 . 2 (𝐴 ∈ V → (𝐴 ∈ ran 𝐹 ↔ (𝐹 “ {𝐴}) ≠ ∅))
201, 7, 19pm5.21nii 383 1 (𝐴 ∈ ran 𝐹 ↔ (𝐹 “ {𝐴}) ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2735   ≠ wne 2951  Vcvv 3409  ∅c0 4227  {csn 4525   class class class wbr 5035  ◡ccnv 5526  ran crn 5528   “ cima 5530 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5036  df-opab 5098  df-xp 5533  df-cnv 5535  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540 This theorem is referenced by:  fnpreimac  30536  dnnumch3lem  40391  dnnumch3  40392  wessf1ornlem  42209
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