Theorem inisegn0 6065

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 3463	. 2 ⊢ (𝐴 ∈ ran 𝐹 → 𝐴 ∈ V)
2		snprc 4676	. . . . . 6 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3	2	biimpi 216	. . . . 5 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
4	3	imaeq2d 6027	. . . 4 ⊢ (¬ 𝐴 ∈ V → (◡𝐹 “ {𝐴}) = (◡𝐹 “ ∅))
5		ima0 6044	. . . 4 ⊢ (◡𝐹 “ ∅) = ∅
6	4, 5	eqtrdi 2788	. . 3 ⊢ (¬ 𝐴 ∈ V → (◡𝐹 “ {𝐴}) = ∅)
7	6	necon1ai 2960	. 2 ⊢ ((◡𝐹 “ {𝐴}) ≠ ∅ → 𝐴 ∈ V)
8		eleq1 2825	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ ran 𝐹 ↔ 𝐴 ∈ ran 𝐹))
9		sneq 4592	. . . . 5 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
10	9	imaeq2d 6027	. . . 4 ⊢ (𝑎 = 𝐴 → (◡𝐹 “ {𝑎}) = (◡𝐹 “ {𝐴}))
11	10	neeq1d 2992	. . 3 ⊢ (𝑎 = 𝐴 → ((◡𝐹 “ {𝑎}) ≠ ∅ ↔ (◡𝐹 “ {𝐴}) ≠ ∅))
12		abn0 4339	. . . 4 ⊢ ({𝑏 ∣ 𝑏𝐹𝑎} ≠ ∅ ↔ ∃𝑏 𝑏𝐹𝑎)
13		iniseg 6064	. . . . . 6 ⊢ (𝑎 ∈ V → (◡𝐹 “ {𝑎}) = {𝑏 ∣ 𝑏𝐹𝑎})
14	13	elv 3447	. . . . 5 ⊢ (◡𝐹 “ {𝑎}) = {𝑏 ∣ 𝑏𝐹𝑎}
15	14	neeq1i 2997	. . . 4 ⊢ ((◡𝐹 “ {𝑎}) ≠ ∅ ↔ {𝑏 ∣ 𝑏𝐹𝑎} ≠ ∅)
16		vex 3446	. . . . 5 ⊢ 𝑎 ∈ V
17	16	elrn 5850	. . . 4 ⊢ (𝑎 ∈ ran 𝐹 ↔ ∃𝑏 𝑏𝐹𝑎)
18	12, 15, 17	3bitr4ri 304	. . 3 ⊢ (𝑎 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑎}) ≠ ∅)
19	8, 11, 18	vtoclbg 3516	. 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 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715   ≠ wne 2933  Vcvv 3442  ∅c0 4287  {csn 4582   class class class wbr 5100  ^◡ccnv 5631  ran crn 5633   “ cima 5635

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645

This theorem is referenced by:  fnpreimac  32760  dnnumch3lem  43403  dnnumch3  43404  wessf1ornlem  45544