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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 3512 | . 2 ⊢ (𝐴 ∈ ran 𝐹 → 𝐴 ∈ V) | |
2 | snprc 4653 | . . . . . 6 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
3 | 2 | biimpi 218 | . . . . 5 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
4 | 3 | imaeq2d 5929 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (◡𝐹 “ {𝐴}) = (◡𝐹 “ ∅)) |
5 | ima0 5945 | . . . 4 ⊢ (◡𝐹 “ ∅) = ∅ | |
6 | 4, 5 | syl6eq 2872 | . . 3 ⊢ (¬ 𝐴 ∈ V → (◡𝐹 “ {𝐴}) = ∅) |
7 | 6 | necon1ai 3043 | . 2 ⊢ ((◡𝐹 “ {𝐴}) ≠ ∅ → 𝐴 ∈ V) |
8 | eleq1 2900 | . . 3 ⊢ (𝑎 = 𝐴 → (𝑎 ∈ ran 𝐹 ↔ 𝐴 ∈ ran 𝐹)) | |
9 | sneq 4577 | . . . . 5 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴}) | |
10 | 9 | imaeq2d 5929 | . . . 4 ⊢ (𝑎 = 𝐴 → (◡𝐹 “ {𝑎}) = (◡𝐹 “ {𝐴})) |
11 | 10 | neeq1d 3075 | . . 3 ⊢ (𝑎 = 𝐴 → ((◡𝐹 “ {𝑎}) ≠ ∅ ↔ (◡𝐹 “ {𝐴}) ≠ ∅)) |
12 | abn0 4336 | . . . 4 ⊢ ({𝑏 ∣ 𝑏𝐹𝑎} ≠ ∅ ↔ ∃𝑏 𝑏𝐹𝑎) | |
13 | iniseg 5960 | . . . . . 6 ⊢ (𝑎 ∈ V → (◡𝐹 “ {𝑎}) = {𝑏 ∣ 𝑏𝐹𝑎}) | |
14 | 13 | elv 3499 | . . . . 5 ⊢ (◡𝐹 “ {𝑎}) = {𝑏 ∣ 𝑏𝐹𝑎} |
15 | 14 | neeq1i 3080 | . . . 4 ⊢ ((◡𝐹 “ {𝑎}) ≠ ∅ ↔ {𝑏 ∣ 𝑏𝐹𝑎} ≠ ∅) |
16 | vex 3497 | . . . . 5 ⊢ 𝑎 ∈ V | |
17 | 16 | elrn 5822 | . . . 4 ⊢ (𝑎 ∈ ran 𝐹 ↔ ∃𝑏 𝑏𝐹𝑎) |
18 | 12, 15, 17 | 3bitr4ri 306 | . . 3 ⊢ (𝑎 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑎}) ≠ ∅) |
19 | 8, 11, 18 | vtoclbg 3569 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝐴}) ≠ ∅)) |
20 | 1, 7, 19 | pm5.21nii 382 | 1 ⊢ (𝐴 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝐴}) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {cab 2799 ≠ wne 3016 Vcvv 3494 ∅c0 4291 {csn 4567 class class class wbr 5066 ◡ccnv 5554 ran crn 5556 “ cima 5558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 |
This theorem is referenced by: fnpreimac 30416 dnnumch3lem 39666 dnnumch3 39667 wessf1ornlem 41465 |
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