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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 3493 | . 2 ⊢ (𝐴 ∈ ran 𝐹 → 𝐴 ∈ V) | |
2 | snprc 4720 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
3 | 2 | biimpi 215 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
4 | 3 | imaeq2d 6057 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (◡𝐹 “ {𝐴}) = (◡𝐹 “ ∅)) |
5 | ima0 6073 | . . . 4 ⊢ (◡𝐹 “ ∅) = ∅ | |
6 | 4, 5 | eqtrdi 2789 | . . 3 ⊢ (¬ 𝐴 ∈ V → (◡𝐹 “ {𝐴}) = ∅) |
7 | 6 | necon1ai 2969 | . 2 ⊢ ((◡𝐹 “ {𝐴}) ≠ ∅ → 𝐴 ∈ V) |
8 | eleq1 2822 | . . 3 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ ran 𝐹 ↔ 𝐴 ∈ ran 𝐹)) | |
9 | sneq 4637 | . . . . 5 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴}) | |
10 | 9 | imaeq2d 6057 | . . . 4 ⊢ (𝑎 = 𝐴 → (◡𝐹 “ {𝑎}) = (◡𝐹 “ {𝐴})) |
11 | 10 | neeq1d 3001 | . . 3 ⊢ (𝑎 = 𝐴 → ((◡𝐹 “ {𝑎}) ≠ ∅ ↔ (◡𝐹 “ {𝐴}) ≠ ∅)) |
12 | abn0 4379 | . . . 4 ⊢ ({𝑏 ∣ 𝑏𝐹𝑎} ≠ ∅ ↔ ∃𝑏 𝑏𝐹𝑎) | |
13 | iniseg 6093 | . . . . . 6 ⊢ (𝑎 ∈ V → (◡𝐹 “ {𝑎}) = {𝑏 ∣ 𝑏𝐹𝑎}) | |
14 | 13 | elv 3481 | . . . . 5 ⊢ (◡𝐹 “ {𝑎}) = {𝑏 ∣ 𝑏𝐹𝑎} |
15 | 14 | neeq1i 3006 | . . . 4 ⊢ ((◡𝐹 “ {𝑎}) ≠ ∅ ↔ {𝑏 ∣ 𝑏𝐹𝑎} ≠ ∅) |
16 | vex 3479 | . . . . 5 ⊢ 𝑎 ∈ V | |
17 | 16 | elrn 5891 | . . . 4 ⊢ (𝑎 ∈ ran 𝐹 ↔ ∃𝑏 𝑏𝐹𝑎) |
18 | 12, 15, 17 | 3bitr4ri 304 | . . 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 380 | 1 ⊢ (𝐴 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝐴}) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1542 ∃wex 1782 ∈ wcel 2107 {cab 2710 ≠ wne 2941 Vcvv 3475 ∅c0 4321 {csn 4627 class class class wbr 5147 ◡ccnv 5674 ran crn 5676 “ cima 5678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-xp 5681 df-cnv 5683 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 |
This theorem is referenced by: fnpreimac 31874 dnnumch3lem 41721 dnnumch3 41722 wessf1ornlem 43815 |
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