![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 3499 | . 2 ⊢ (𝐴 ∈ ran 𝐹 → 𝐴 ∈ V) | |
2 | snprc 4722 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
3 | 2 | biimpi 216 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
4 | 3 | imaeq2d 6080 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (◡𝐹 “ {𝐴}) = (◡𝐹 “ ∅)) |
5 | ima0 6097 | . . . 4 ⊢ (◡𝐹 “ ∅) = ∅ | |
6 | 4, 5 | eqtrdi 2791 | . . 3 ⊢ (¬ 𝐴 ∈ V → (◡𝐹 “ {𝐴}) = ∅) |
7 | 6 | necon1ai 2966 | . 2 ⊢ ((◡𝐹 “ {𝐴}) ≠ ∅ → 𝐴 ∈ V) |
8 | eleq1 2827 | . . 3 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ ran 𝐹 ↔ 𝐴 ∈ ran 𝐹)) | |
9 | sneq 4641 | . . . . 5 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴}) | |
10 | 9 | imaeq2d 6080 | . . . 4 ⊢ (𝑎 = 𝐴 → (◡𝐹 “ {𝑎}) = (◡𝐹 “ {𝐴})) |
11 | 10 | neeq1d 2998 | . . 3 ⊢ (𝑎 = 𝐴 → ((◡𝐹 “ {𝑎}) ≠ ∅ ↔ (◡𝐹 “ {𝐴}) ≠ ∅)) |
12 | abn0 4391 | . . . 4 ⊢ ({𝑏 ∣ 𝑏𝐹𝑎} ≠ ∅ ↔ ∃𝑏 𝑏𝐹𝑎) | |
13 | iniseg 6118 | . . . . . 6 ⊢ (𝑎 ∈ V → (◡𝐹 “ {𝑎}) = {𝑏 ∣ 𝑏𝐹𝑎}) | |
14 | 13 | elv 3483 | . . . . 5 ⊢ (◡𝐹 “ {𝑎}) = {𝑏 ∣ 𝑏𝐹𝑎} |
15 | 14 | neeq1i 3003 | . . . 4 ⊢ ((◡𝐹 “ {𝑎}) ≠ ∅ ↔ {𝑏 ∣ 𝑏𝐹𝑎} ≠ ∅) |
16 | vex 3482 | . . . . 5 ⊢ 𝑎 ∈ V | |
17 | 16 | elrn 5907 | . . . 4 ⊢ (𝑎 ∈ ran 𝐹 ↔ ∃𝑏 𝑏𝐹𝑎) |
18 | 12, 15, 17 | 3bitr4ri 304 | . . 3 ⊢ (𝑎 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑎}) ≠ ∅) |
19 | 8, 11, 18 | vtoclbg 3557 | . 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 1537 ∃wex 1776 ∈ wcel 2106 {cab 2712 ≠ wne 2938 Vcvv 3478 ∅c0 4339 {csn 4631 class class class wbr 5148 ◡ccnv 5688 ran crn 5690 “ cima 5692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 |
This theorem is referenced by: fnpreimac 32688 dnnumch3lem 43035 dnnumch3 43036 wessf1ornlem 45128 |
Copyright terms: Public domain | W3C validator |