Theorem inisegn0 6006

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.

Step	Hyp	Ref	Expression
1		elex 3450	. 2 ⊢ (𝐴 ∈ ran 𝐹 → 𝐴 ∈ V)
2		snprc 4653	. . . . . 6 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3	2	biimpi 215	. . . . 5 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
4	3	imaeq2d 5969	. . . 4 ⊢ (¬ 𝐴 ∈ V → (◡𝐹 “ {𝐴}) = (◡𝐹 “ ∅))
5		ima0 5985	. . . 4 ⊢ (◡𝐹 “ ∅) = ∅
6	4, 5	eqtrdi 2794	. . 3 ⊢ (¬ 𝐴 ∈ V → (◡𝐹 “ {𝐴}) = ∅)
7	6	necon1ai 2971	. 2 ⊢ ((◡𝐹 “ {𝐴}) ≠ ∅ → 𝐴 ∈ V)
8		eleq1 2826	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ ran 𝐹 ↔ 𝐴 ∈ ran 𝐹))
9		sneq 4571	. . . . 5 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
10	9	imaeq2d 5969	. . . 4 ⊢ (𝑎 = 𝐴 → (◡𝐹 “ {𝑎}) = (◡𝐹 “ {𝐴}))
11	10	neeq1d 3003	. . 3 ⊢ (𝑎 = 𝐴 → ((◡𝐹 “ {𝑎}) ≠ ∅ ↔ (◡𝐹 “ {𝐴}) ≠ ∅))
12		abn0 4314	. . . 4 ⊢ ({𝑏 ∣ 𝑏𝐹𝑎} ≠ ∅ ↔ ∃𝑏 𝑏𝐹𝑎)
13		iniseg 6005	. . . . . 6 ⊢ (𝑎 ∈ V → (◡𝐹 “ {𝑎}) = {𝑏 ∣ 𝑏𝐹𝑎})
14	13	elv 3438	. . . . 5 ⊢ (◡𝐹 “ {𝑎}) = {𝑏 ∣ 𝑏𝐹𝑎}
15	14	neeq1i 3008	. . . 4 ⊢ ((◡𝐹 “ {𝑎}) ≠ ∅ ↔ {𝑏 ∣ 𝑏𝐹𝑎} ≠ ∅)
16		vex 3436	. . . . 5 ⊢ 𝑎 ∈ V
17	16	elrn 5802	. . . 4 ⊢ (𝑎 ∈ ran 𝐹 ↔ ∃𝑏 𝑏𝐹𝑎)
18	12, 15, 17	3bitr4ri 304	. . 3 ⊢ (𝑎 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑎}) ≠ ∅)
19	8, 11, 18	vtoclbg 3507	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝐴}) ≠ ∅))
20	1, 7, 19	pm5.21nii 380	1 ⊢ (𝐴 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝐴}) ≠ ∅)

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715   ≠ wne 2943  Vcvv 3432  ∅c0 4256  {csn 4561   class class class wbr 5074  ^◡ccnv 5588  ran crn 5590   “ cima 5592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602

This theorem is referenced by:  fnpreimac  31008  dnnumch3lem  40871  dnnumch3  40872  wessf1ornlem  42722