Theorem inisegn0 6050

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 3452	. 2 ⊢ (𝐴 ∈ ran 𝐹 → 𝐴 ∈ V)
2		snprc 4649	. . . . . 6 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3	2	biimpi 217	. . . . 5 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
4	3	imaeq2d 6012	. . . 4 ⊢ (¬ 𝐴 ∈ V → (◡𝐹 “ {𝐴}) = (◡𝐹 “ ∅))
5		ima0 6029	. . . 4 ⊢ (◡𝐹 “ ∅) = ∅
6	4, 5	eqtrdi 2790	. . 3 ⊢ (¬ 𝐴 ∈ V → (◡𝐹 “ {𝐴}) = ∅)
7	6	necon1ai 2961	. 2 ⊢ ((◡𝐹 “ {𝐴}) ≠ ∅ → 𝐴 ∈ V)
8		eleq1 2827	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ ran 𝐹 ↔ 𝐴 ∈ ran 𝐹))
9		sneq 4565	. . . . 5 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
10	9	imaeq2d 6012	. . . 4 ⊢ (𝑎 = 𝐴 → (◡𝐹 “ {𝑎}) = (◡𝐹 “ {𝐴}))
11	10	neeq1d 2993	. . 3 ⊢ (𝑎 = 𝐴 → ((◡𝐹 “ {𝑎}) ≠ ∅ ↔ (◡𝐹 “ {𝐴}) ≠ ∅))
12		abn0 4313	. . . 4 ⊢ ({𝑏 ∣ 𝑏𝐹𝑎} ≠ ∅ ↔ ∃𝑏 𝑏𝐹𝑎)
13		iniseg 6049	. . . . . 6 ⊢ (𝑎 ∈ V → (◡𝐹 “ {𝑎}) = {𝑏 ∣ 𝑏𝐹𝑎})
14	13	elv 3436	. . . . 5 ⊢ (◡𝐹 “ {𝑎}) = {𝑏 ∣ 𝑏𝐹𝑎}
15	14	neeq1i 2998	. . . 4 ⊢ ((◡𝐹 “ {𝑎}) ≠ ∅ ↔ {𝑏 ∣ 𝑏𝐹𝑎} ≠ ∅)
16		vex 3435	. . . . 5 ⊢ 𝑎 ∈ V
17	16	elrn 5835	. . . 4 ⊢ (𝑎 ∈ ran 𝐹 ↔ ∃𝑏 𝑏𝐹𝑎)
18	12, 15, 17	3bitr4ri 305	. . 3 ⊢ (𝑎 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑎}) ≠ ∅)
19	8, 11, 18	vtoclbg 3502	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝐴}) ≠ ∅))
20	1, 7, 19	pm5.21nii 379	1 ⊢ (𝐴 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝐴}) ≠ ∅)

Syntax hints:  ¬ wn 3   ↔ wb 207   = wceq 1547  ∃wex 1786   ∈ wcel 2119  {cab 2717   ≠ wne 2934  Vcvv 3431  ∅c0 4261  {csn 4555   class class class wbr 5072  ^◡ccnv 5617  ran crn 5619   “ cima 5621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-xp 5624  df-cnv 5626  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631

This theorem is referenced by:  fnpreimac  32762  dnnumch3lem  43491  dnnumch3  43492  wessf1ornlem  45632