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Mirrors > Home > MPE Home > Th. List > fnelnfp | Structured version Visualization version GIF version |
Description: Property of a non-fixed point of a function. (Contributed by Stefan O'Rear, 15-Aug-2015.) |
Ref | Expression |
---|---|
fnelnfp | ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑋 ∈ dom (𝐹 ∖ I ) ↔ (𝐹‘𝑋) ≠ 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fndifnfp 7121 | . . 3 ⊢ (𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥}) | |
2 | 1 | eleq2d 2823 | . 2 ⊢ (𝐹 Fn 𝐴 → (𝑋 ∈ dom (𝐹 ∖ I ) ↔ 𝑋 ∈ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ 𝑥})) |
3 | fveq2 6842 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
4 | id 22 | . . . 4 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) | |
5 | 3, 4 | neeq12d 3005 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ≠ 𝑥 ↔ (𝐹‘𝑋) ≠ 𝑋)) |
6 | 5 | elrab3 3646 | . 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 ≠ wne 2943 {crab 3407 ∖ cdif 3907 I cid 5530 dom cdm 5633 Fn wfn 6491 ‘cfv 6496 |
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-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-fv 6504 |
This theorem is referenced by: f1omvdmvd 19223 f1omvdconj 19226 f1otrspeq 19227 pmtrfinv 19241 symggen 19250 psgnunilem1 19273 mdetdiaglem 21945 mdetralt 21955 mdetunilem7 21965 nfpconfp 31493 pmtrcnel 31884 pmtrcnel2 31885 pmtrcnelor 31886 cycpmrn 31936 |
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