Theorem inisegn0 6090

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 3485	. 2 ⊢ (𝐴 ∈ ran 𝐹 → 𝐴 ∈ V)
2		snprc 4698	. . . . . 6 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3	2	biimpi 216	. . . . 5 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
4	3	imaeq2d 6052	. . . 4 ⊢ (¬ 𝐴 ∈ V → (◡𝐹 “ {𝐴}) = (◡𝐹 “ ∅))
5		ima0 6069	. . . 4 ⊢ (◡𝐹 “ ∅) = ∅
6	4, 5	eqtrdi 2787	. . 3 ⊢ (¬ 𝐴 ∈ V → (◡𝐹 “ {𝐴}) = ∅)
7	6	necon1ai 2960	. 2 ⊢ ((◡𝐹 “ {𝐴}) ≠ ∅ → 𝐴 ∈ V)
8		eleq1 2823	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ ran 𝐹 ↔ 𝐴 ∈ ran 𝐹))
9		sneq 4616	. . . . 5 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
10	9	imaeq2d 6052	. . . 4 ⊢ (𝑎 = 𝐴 → (◡𝐹 “ {𝑎}) = (◡𝐹 “ {𝐴}))
11	10	neeq1d 2992	. . 3 ⊢ (𝑎 = 𝐴 → ((◡𝐹 “ {𝑎}) ≠ ∅ ↔ (◡𝐹 “ {𝐴}) ≠ ∅))
12		abn0 4365	. . . 4 ⊢ ({𝑏 ∣ 𝑏𝐹𝑎} ≠ ∅ ↔ ∃𝑏 𝑏𝐹𝑎)
13		iniseg 6089	. . . . . 6 ⊢ (𝑎 ∈ V → (◡𝐹 “ {𝑎}) = {𝑏 ∣ 𝑏𝐹𝑎})
14	13	elv 3469	. . . . 5 ⊢ (◡𝐹 “ {𝑎}) = {𝑏 ∣ 𝑏𝐹𝑎}
15	14	neeq1i 2997	. . . 4 ⊢ ((◡𝐹 “ {𝑎}) ≠ ∅ ↔ {𝑏 ∣ 𝑏𝐹𝑎} ≠ ∅)
16		vex 3468	. . . . 5 ⊢ 𝑎 ∈ V
17	16	elrn 5878	. . . 4 ⊢ (𝑎 ∈ ran 𝐹 ↔ ∃𝑏 𝑏𝐹𝑎)
18	12, 15, 17	3bitr4ri 304	. . 3 ⊢ (𝑎 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑎}) ≠ ∅)
19	8, 11, 18	vtoclbg 3541	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝐴}) ≠ ∅))
20	1, 7, 19	pm5.21nii 378	1 ⊢ (𝐴 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝐴}) ≠ ∅)

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2714   ≠ wne 2933  Vcvv 3464  ∅c0 4313  {csn 4606   class class class wbr 5124  ^◡ccnv 5658  ran crn 5660   “ cima 5662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672

This theorem is referenced by:  fnpreimac  32654  dnnumch3lem  43037  dnnumch3  43038  wessf1ornlem  45176