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| Description: Nonemptiness of an initial segment in terms of range. (Contributed by Stefan O'Rear, 18-Jan-2015.) | 
| Ref | Expression | 
|---|---|
| inisegn0 | ⊢ (𝐴 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝐴}) ≠ ∅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elex 3500 | . 2 ⊢ (𝐴 ∈ ran 𝐹 → 𝐴 ∈ V) | |
| 2 | snprc 4716 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
| 3 | 2 | biimpi 216 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) | 
| 4 | 3 | imaeq2d 6077 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (◡𝐹 “ {𝐴}) = (◡𝐹 “ ∅)) | 
| 5 | ima0 6094 | . . . 4 ⊢ (◡𝐹 “ ∅) = ∅ | |
| 6 | 4, 5 | eqtrdi 2792 | . . 3 ⊢ (¬ 𝐴 ∈ V → (◡𝐹 “ {𝐴}) = ∅) | 
| 7 | 6 | necon1ai 2967 | . 2 ⊢ ((◡𝐹 “ {𝐴}) ≠ ∅ → 𝐴 ∈ V) | 
| 8 | eleq1 2828 | . . 3 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ ran 𝐹 ↔ 𝐴 ∈ ran 𝐹)) | |
| 9 | sneq 4635 | . . . . 5 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴}) | |
| 10 | 9 | imaeq2d 6077 | . . . 4 ⊢ (𝑎 = 𝐴 → (◡𝐹 “ {𝑎}) = (◡𝐹 “ {𝐴})) | 
| 11 | 10 | neeq1d 2999 | . . 3 ⊢ (𝑎 = 𝐴 → ((◡𝐹 “ {𝑎}) ≠ ∅ ↔ (◡𝐹 “ {𝐴}) ≠ ∅)) | 
| 12 | abn0 4384 | . . . 4 ⊢ ({𝑏 ∣ 𝑏𝐹𝑎} ≠ ∅ ↔ ∃𝑏 𝑏𝐹𝑎) | |
| 13 | iniseg 6114 | . . . . . 6 ⊢ (𝑎 ∈ V → (◡𝐹 “ {𝑎}) = {𝑏 ∣ 𝑏𝐹𝑎}) | |
| 14 | 13 | elv 3484 | . . . . 5 ⊢ (◡𝐹 “ {𝑎}) = {𝑏 ∣ 𝑏𝐹𝑎} | 
| 15 | 14 | neeq1i 3004 | . . . 4 ⊢ ((◡𝐹 “ {𝑎}) ≠ ∅ ↔ {𝑏 ∣ 𝑏𝐹𝑎} ≠ ∅) | 
| 16 | vex 3483 | . . . . 5 ⊢ 𝑎 ∈ V | |
| 17 | 16 | elrn 5903 | . . . 4 ⊢ (𝑎 ∈ ran 𝐹 ↔ ∃𝑏 𝑏𝐹𝑎) | 
| 18 | 12, 15, 17 | 3bitr4ri 304 | . . 3 ⊢ (𝑎 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑎}) ≠ ∅) | 
| 19 | 8, 11, 18 | vtoclbg 3556 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝐴}) ≠ ∅)) | 
| 20 | 1, 7, 19 | pm5.21nii 378 | 1 ⊢ (𝐴 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝐴}) ≠ ∅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ↔ wb 206 = wceq 1539 ∃wex 1778 ∈ wcel 2107 {cab 2713 ≠ wne 2939 Vcvv 3479 ∅c0 4332 {csn 4625 class class class wbr 5142 ◡ccnv 5683 ran crn 5685 “ cima 5687 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-cnv 5692 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 | 
| This theorem is referenced by: fnpreimac 32682 dnnumch3lem 43063 dnnumch3 43064 wessf1ornlem 45195 | 
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