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Mirrors > Home > MPE Home > Th. List > inisegn0 | Structured version Visualization version GIF version |
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 3492 | . 2 ⊢ (𝐴 ∈ ran 𝐹 → 𝐴 ∈ V) | |
2 | snprc 4721 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
3 | 2 | biimpi 215 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
4 | 3 | imaeq2d 6059 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (◡𝐹 “ {𝐴}) = (◡𝐹 “ ∅)) |
5 | ima0 6076 | . . . 4 ⊢ (◡𝐹 “ ∅) = ∅ | |
6 | 4, 5 | eqtrdi 2788 | . . 3 ⊢ (¬ 𝐴 ∈ V → (◡𝐹 “ {𝐴}) = ∅) |
7 | 6 | necon1ai 2968 | . 2 ⊢ ((◡𝐹 “ {𝐴}) ≠ ∅ → 𝐴 ∈ V) |
8 | eleq1 2821 | . . 3 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ ran 𝐹 ↔ 𝐴 ∈ ran 𝐹)) | |
9 | sneq 4638 | . . . . 5 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴}) | |
10 | 9 | imaeq2d 6059 | . . . 4 ⊢ (𝑎 = 𝐴 → (◡𝐹 “ {𝑎}) = (◡𝐹 “ {𝐴})) |
11 | 10 | neeq1d 3000 | . . 3 ⊢ (𝑎 = 𝐴 → ((◡𝐹 “ {𝑎}) ≠ ∅ ↔ (◡𝐹 “ {𝐴}) ≠ ∅)) |
12 | abn0 4380 | . . . 4 ⊢ ({𝑏 ∣ 𝑏𝐹𝑎} ≠ ∅ ↔ ∃𝑏 𝑏𝐹𝑎) | |
13 | iniseg 6096 | . . . . . 6 ⊢ (𝑎 ∈ V → (◡𝐹 “ {𝑎}) = {𝑏 ∣ 𝑏𝐹𝑎}) | |
14 | 13 | elv 3480 | . . . . 5 ⊢ (◡𝐹 “ {𝑎}) = {𝑏 ∣ 𝑏𝐹𝑎} |
15 | 14 | neeq1i 3005 | . . . 4 ⊢ ((◡𝐹 “ {𝑎}) ≠ ∅ ↔ {𝑏 ∣ 𝑏𝐹𝑎} ≠ ∅) |
16 | vex 3478 | . . . . 5 ⊢ 𝑎 ∈ V | |
17 | 16 | elrn 5893 | . . . 4 ⊢ (𝑎 ∈ ran 𝐹 ↔ ∃𝑏 𝑏𝐹𝑎) |
18 | 12, 15, 17 | 3bitr4ri 303 | . . 3 ⊢ (𝑎 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑎}) ≠ ∅) |
19 | 8, 11, 18 | vtoclbg 3559 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝐴}) ≠ ∅)) |
20 | 1, 7, 19 | pm5.21nii 379 | 1 ⊢ (𝐴 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝐴}) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1541 ∃wex 1781 ∈ wcel 2106 {cab 2709 ≠ wne 2940 Vcvv 3474 ∅c0 4322 {csn 4628 class class class wbr 5148 ◡ccnv 5675 ran crn 5677 “ cima 5679 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 |
This theorem is referenced by: fnpreimac 31891 dnnumch3lem 41778 dnnumch3 41779 wessf1ornlem 43872 |
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