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Theorem inisegn0 5653
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 3350 . 2 (𝐴 ∈ ran 𝐹𝐴 ∈ V)
2 snprc 4395 . . . . . 6 𝐴 ∈ V ↔ {𝐴} = ∅)
32biimpi 206 . . . . 5 𝐴 ∈ V → {𝐴} = ∅)
43imaeq2d 5622 . . . 4 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅))
5 ima0 5637 . . . 4 (𝐹 “ ∅) = ∅
64, 5syl6eq 2808 . . 3 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅)
76necon1ai 2957 . 2 ((𝐹 “ {𝐴}) ≠ ∅ → 𝐴 ∈ V)
8 eleq1 2825 . . 3 (𝑎 = 𝐴 → (𝑎 ∈ ran 𝐹𝐴 ∈ ran 𝐹))
9 sneq 4329 . . . . 5 (𝑎 = 𝐴 → {𝑎} = {𝐴})
109imaeq2d 5622 . . . 4 (𝑎 = 𝐴 → (𝐹 “ {𝑎}) = (𝐹 “ {𝐴}))
1110neeq1d 2989 . . 3 (𝑎 = 𝐴 → ((𝐹 “ {𝑎}) ≠ ∅ ↔ (𝐹 “ {𝐴}) ≠ ∅))
12 abn0 4095 . . . 4 ({𝑏𝑏𝐹𝑎} ≠ ∅ ↔ ∃𝑏 𝑏𝐹𝑎)
13 vex 3341 . . . . . 6 𝑎 ∈ V
14 iniseg 5652 . . . . . 6 (𝑎 ∈ V → (𝐹 “ {𝑎}) = {𝑏𝑏𝐹𝑎})
1513, 14ax-mp 5 . . . . 5 (𝐹 “ {𝑎}) = {𝑏𝑏𝐹𝑎}
1615neeq1i 2994 . . . 4 ((𝐹 “ {𝑎}) ≠ ∅ ↔ {𝑏𝑏𝐹𝑎} ≠ ∅)
1713elrn 5519 . . . 4 (𝑎 ∈ ran 𝐹 ↔ ∃𝑏 𝑏𝐹𝑎)
1812, 16, 173bitr4ri 293 . . 3 (𝑎 ∈ ran 𝐹 ↔ (𝐹 “ {𝑎}) ≠ ∅)
198, 11, 18vtoclbg 3405 . 2 (𝐴 ∈ V → (𝐴 ∈ ran 𝐹 ↔ (𝐹 “ {𝐴}) ≠ ∅))
201, 7, 19pm5.21nii 367 1 (𝐴 ∈ ran 𝐹 ↔ (𝐹 “ {𝐴}) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196   = wceq 1630  wex 1851  wcel 2137  {cab 2744  wne 2930  Vcvv 3338  c0 4056  {csn 4319   class class class wbr 4802  ccnv 5263  ran crn 5265  cima 5267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-br 4803  df-opab 4863  df-xp 5270  df-cnv 5272  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277
This theorem is referenced by:  dnnumch3lem  38116  dnnumch3  38117  wessf1ornlem  39868
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