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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 3350 | . 2 ⊢ (𝐴 ∈ ran 𝐹 → 𝐴 ∈ V) | |
2 | snprc 4395 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
3 | 2 | biimpi 206 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
4 | 3 | imaeq2d 5622 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (◡𝐹 “ {𝐴}) = (◡𝐹 “ ∅)) |
5 | ima0 5637 | . . . 4 ⊢ (◡𝐹 “ ∅) = ∅ | |
6 | 4, 5 | syl6eq 2808 | . . 3 ⊢ (¬ 𝐴 ∈ V → (◡𝐹 “ {𝐴}) = ∅) |
7 | 6 | necon1ai 2957 | . 2 ⊢ ((◡𝐹 “ {𝐴}) ≠ ∅ → 𝐴 ∈ V) |
8 | eleq1 2825 | . . 3 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ ran 𝐹 ↔ 𝐴 ∈ ran 𝐹)) | |
9 | sneq 4329 | . . . . 5 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴}) | |
10 | 9 | imaeq2d 5622 | . . . 4 ⊢ (𝑎 = 𝐴 → (◡𝐹 “ {𝑎}) = (◡𝐹 “ {𝐴})) |
11 | 10 | neeq1d 2989 | . . 3 ⊢ (𝑎 = 𝐴 → ((◡𝐹 “ {𝑎}) ≠ ∅ ↔ (◡𝐹 “ {𝐴}) ≠ ∅)) |
12 | abn0 4095 | . . . 4 ⊢ ({𝑏 ∣ 𝑏𝐹𝑎} ≠ ∅ ↔ ∃𝑏 𝑏𝐹𝑎) | |
13 | vex 3341 | . . . . . 6 ⊢ 𝑎 ∈ V | |
14 | iniseg 5652 | . . . . . 6 ⊢ (𝑎 ∈ V → (◡𝐹 “ {𝑎}) = {𝑏 ∣ 𝑏𝐹𝑎}) | |
15 | 13, 14 | ax-mp 5 | . . . . 5 ⊢ (◡𝐹 “ {𝑎}) = {𝑏 ∣ 𝑏𝐹𝑎} |
16 | 15 | neeq1i 2994 | . . . 4 ⊢ ((◡𝐹 “ {𝑎}) ≠ ∅ ↔ {𝑏 ∣ 𝑏𝐹𝑎} ≠ ∅) |
17 | 13 | elrn 5519 | . . . 4 ⊢ (𝑎 ∈ ran 𝐹 ↔ ∃𝑏 𝑏𝐹𝑎) |
18 | 12, 16, 17 | 3bitr4ri 293 | . . 3 ⊢ (𝑎 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑎}) ≠ ∅) |
19 | 8, 11, 18 | vtoclbg 3405 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝐴}) ≠ ∅)) |
20 | 1, 7, 19 | pm5.21nii 367 | 1 ⊢ (𝐴 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝐴}) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 = wceq 1630 ∃wex 1851 ∈ wcel 2137 {cab 2744 ≠ wne 2930 Vcvv 3338 ∅c0 4056 {csn 4319 class class class wbr 4802 ◡ccnv 5263 ran crn 5265 “ cima 5267 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1986 ax-6 2052 ax-7 2088 ax-9 2146 ax-10 2166 ax-11 2181 ax-12 2194 ax-13 2389 ax-ext 2738 ax-sep 4931 ax-nul 4939 ax-pr 5053 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2045 df-eu 2609 df-mo 2610 df-clab 2745 df-cleq 2751 df-clel 2754 df-nfc 2889 df-ne 2931 df-ral 3053 df-rex 3054 df-rab 3057 df-v 3340 df-sbc 3575 df-dif 3716 df-un 3718 df-in 3720 df-ss 3727 df-nul 4057 df-if 4229 df-sn 4320 df-pr 4322 df-op 4326 df-br 4803 df-opab 4863 df-xp 5270 df-cnv 5272 df-dm 5274 df-rn 5275 df-res 5276 df-ima 5277 |
This theorem is referenced by: dnnumch3lem 38116 dnnumch3 38117 wessf1ornlem 39868 |
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