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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfl1lem | Structured version Visualization version GIF version | ||
| Description: Property of a functional with a closed kernel. (Contributed by NM, 28-Dec-2014.) | 
| Ref | Expression | 
|---|---|
| lcfl1.c | ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | 
| Ref | Expression | 
|---|---|
| lcfl1lem | ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fveq2 6906 | . . . . 5 ⊢ (𝑓 = 𝐺 → (𝐿‘𝑓) = (𝐿‘𝐺)) | |
| 2 | 1 | fveq2d 6910 | . . . 4 ⊢ (𝑓 = 𝐺 → ( ⊥ ‘(𝐿‘𝑓)) = ( ⊥ ‘(𝐿‘𝐺))) | 
| 3 | 2 | fveq2d 6910 | . . 3 ⊢ (𝑓 = 𝐺 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺)))) | 
| 4 | 3, 1 | eqeq12d 2753 | . 2 ⊢ (𝑓 = 𝐺 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓) ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))) | 
| 5 | lcfl1.c | . 2 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} | |
| 6 | 4, 5 | elrab2 3695 | 1 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 ‘cfv 6561 | 
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 | 
| This theorem is referenced by: lcfl1 41494 lcfl8b 41506 lclkrlem1 41508 lclkrlem2 41534 lclkr 41535 lcfls1c 41538 lcfrlem9 41552 mapdvalc 41631 mapdval2N 41632 mapdval4N 41634 mapdordlem1a 41636 mapdordlem1bN 41637 mapdrvallem2 41647 | 
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