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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfl1 | Structured version Visualization version GIF version |
Description: Property of a functional with a closed kernel. (Contributed by NM, 31-Dec-2014.) |
Ref | Expression |
---|---|
lcfl1.c | ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} |
lcfl1.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
Ref | Expression |
---|---|
lcfl1 | ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfl1.c | . . 3 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | |
2 | 1 | lcfl1lem 40665 | . 2 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))) |
3 | lcfl1.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
4 | 3 | biantrurd 531 | . 2 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))) |
5 | 2, 4 | bitr4id 289 | 1 ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 {crab 3430 ‘cfv 6542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-rab 3431 df-v 3474 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-uni 4908 df-br 5148 df-iota 6494 df-fv 6550 |
This theorem is referenced by: lcfl2 40667 lcfl5 40670 lcfl5a 40671 lcfl6 40674 lcfl8 40676 lcfl8a 40677 lclkrlem2 40706 |
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