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Mirrors > Home > MPE Home > Th. List > fnelfp | Structured version Visualization version GIF version |
Description: Property of a fixed point of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
Ref | Expression |
---|---|
fnelfp | ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑋 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑋) = 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fninfp 7147 | . . 3 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝑥}) | |
2 | 1 | eleq2d 2818 | . 2 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ dom (𝐹 ∩ I ) ↔ 𝑋 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝑥})) |
3 | fveq2 6869 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
4 | id 22 | . . . 4 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) | |
5 | 3, 4 | eqeq12d 2747 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) = 𝑥 ↔ (𝐹‘𝑋) = 𝑋)) |
6 | 5 | elrab3 3671 | . 2 ⊢ (𝑋 ∈ 𝐴 → (𝑋 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝑥} ↔ (𝐹‘𝑋) = 𝑋)) |
7 | 2, 6 | sylan9bb 510 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑋 ∈ dom (𝐹 ∩ I ) ↔ (𝐹‘𝑋) = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {crab 3425 ∩ cin 3934 I cid 5557 dom cdm 5660 Fn wfn 6518 ‘cfv 6523 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5283 ax-nul 5290 ax-pr 5411 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3426 df-v 3468 df-dif 3938 df-un 3940 df-in 3942 df-ss 3952 df-nul 4310 df-if 4514 df-sn 4614 df-pr 4616 df-op 4620 df-uni 4893 df-br 5133 df-opab 5195 df-mpt 5216 df-id 5558 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-res 5672 df-iota 6475 df-fun 6525 df-fn 6526 df-fv 6531 |
This theorem is referenced by: nfpconfp 31654 ismrcd1 41119 ismrcd2 41120 istopclsd 41121 |
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