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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 7038 | . . 3 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∩ I ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝑥}) | |
2 | 1 | eleq2d 2824 | . 2 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ dom (𝐹 ∩ I ) ↔ 𝑋 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝑥})) |
3 | fveq2 6766 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
4 | id 22 | . . . 4 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) | |
5 | 3, 4 | eqeq12d 2754 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) = 𝑥 ↔ (𝐹‘𝑋) = 𝑋)) |
6 | 5 | elrab3 3624 | . 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 1539 ∈ wcel 2106 {crab 3068 ∩ cin 3885 I cid 5483 dom cdm 5584 Fn wfn 6421 ‘cfv 6426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5221 ax-nul 5228 ax-pr 5350 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3431 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5074 df-opab 5136 df-mpt 5157 df-id 5484 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-res 5596 df-iota 6384 df-fun 6428 df-fn 6429 df-fv 6434 |
This theorem is referenced by: nfpconfp 30975 ismrcd1 40528 ismrcd2 40529 istopclsd 40530 |
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