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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 6663 | . . . . 5 ⊢ (𝑓 = 𝐺 → (𝐿‘𝑓) = (𝐿‘𝐺)) | |
2 | 1 | fveq2d 6667 | . . . 4 ⊢ (𝑓 = 𝐺 → ( ⊥ ‘(𝐿‘𝑓)) = ( ⊥ ‘(𝐿‘𝐺))) |
3 | 2 | fveq2d 6667 | . . 3 ⊢ (𝑓 = 𝐺 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺)))) |
4 | 3, 1 | eqeq12d 2834 | . 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 3680 | 1 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {crab 3139 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 |
This theorem is referenced by: lcfl1 38508 lcfl8b 38520 lclkrlem1 38522 lclkrlem2 38548 lclkr 38549 lcfls1c 38552 lcfrlem9 38566 mapdvalc 38645 mapdval2N 38646 mapdval4N 38648 mapdordlem1a 38650 mapdordlem1bN 38651 mapdrvallem2 38661 |
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