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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 6191 | . . . . 5 ⊢ (𝑓 = 𝐺 → (𝐿‘𝑓) = (𝐿‘𝐺)) | |
2 | 1 | fveq2d 6195 | . . . 4 ⊢ (𝑓 = 𝐺 → ( ⊥ ‘(𝐿‘𝑓)) = ( ⊥ ‘(𝐿‘𝐺))) |
3 | 2 | fveq2d 6195 | . . 3 ⊢ (𝑓 = 𝐺 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺)))) |
4 | 3, 1 | eqeq12d 2637 | . 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 3366 | 1 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 {crab 2916 ‘cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 |
This theorem is referenced by: lcfl1 36781 lcfl8b 36793 lclkrlem1 36795 lclkrlem2 36821 lclkr 36822 lcfls1c 36825 lcfrlem9 36839 mapdvalc 36918 mapdval2N 36919 mapdval4N 36921 mapdordlem1a 36923 mapdordlem1bN 36924 mapdrvallem2 36934 |
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