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Mirrors > Home > MPE Home > Th. List > fniniseg | Structured version Visualization version GIF version |
Description: Membership in the preimage of a singleton, under a function. (Contributed by Mario Carneiro, 12-May-2014.) (Proof shortened by Mario Carneiro , 28-Apr-2015.) |
Ref | Expression |
---|---|
fniniseg | ⊢ (𝐹 Fn 𝐴 → (𝐶 ∈ (◡𝐹 “ {𝐵}) ↔ (𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) = 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpreima 7070 | . 2 ⊢ (𝐹 Fn 𝐴 → (𝐶 ∈ (◡𝐹 “ {𝐵}) ↔ (𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) ∈ {𝐵}))) | |
2 | fvex 6913 | . . . 4 ⊢ (𝐹‘𝐶) ∈ V | |
3 | 2 | elsn 4647 | . . 3 ⊢ ((𝐹‘𝐶) ∈ {𝐵} ↔ (𝐹‘𝐶) = 𝐵) |
4 | 3 | anbi2i 621 | . 2 ⊢ ((𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) ∈ {𝐵}) ↔ (𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) = 𝐵)) |
5 | 1, 4 | bitrdi 286 | 1 ⊢ (𝐹 Fn 𝐴 → (𝐶 ∈ (◡𝐹 “ {𝐵}) ↔ (𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) = 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 {csn 4632 ◡ccnv 5680 “ cima 5684 Fn wfn 6548 ‘cfv 6553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pr 5432 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4325 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5579 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-iota 6505 df-fun 6555 df-fn 6556 df-fv 6561 |
This theorem is referenced by: fparlem1 8125 fparlem2 8126 pw2f1olem 9113 recmulnq 11003 dmrecnq 11007 vdwlem1 16978 vdwlem2 16979 vdwlem6 16983 vdwlem8 16985 vdwlem9 16986 vdwlem12 16989 vdwlem13 16990 ramval 17005 ramub1lem1 17023 ghmeqker 19232 ghmqusnsglem1 19269 ghmquskerlem1 19272 ghmqusker 19276 efgrelexlemb 19743 efgredeu 19745 psgnevpmb 21575 qtopeu 23703 itg1addlem1 25704 i1faddlem 25705 i1fmullem 25706 i1fmulclem 25715 i1fres 25718 itg10a 25723 itg1ge0a 25724 itg1climres 25727 mbfi1fseqlem4 25731 ply1remlem 26184 ply1rem 26185 fta1glem1 26187 fta1glem2 26188 fta1g 26189 fta1blem 26190 plyco0 26211 ofmulrt 26301 plyremlem 26324 plyrem 26325 fta1lem 26327 fta1 26328 vieta1lem1 26330 vieta1lem2 26331 vieta1 26332 plyexmo 26333 elaa 26336 aannenlem1 26348 aalioulem2 26353 pilem1 26473 efif1olem3 26563 efif1olem4 26564 efifo 26566 eff1olem 26567 basellem4 27104 lgsqrlem2 27368 lgsqrlem3 27369 rpvmasum2 27533 dirith 27550 foresf1o 32421 ofpreima 32573 fnpreimac 32579 1stpreimas 32608 s3clhash 32800 pwrssmgc 32858 cycpmconjslem2 33010 cyc3conja 33012 dimkerim 33494 elirng 33533 irngss 33534 irngnzply1 33538 locfinreflem 33611 qqhre 33791 indpi1 33809 indpreima 33814 sibfof 34130 cvmliftlem6 35070 cvmliftlem7 35071 cvmliftlem8 35072 cvmliftlem9 35073 taupilem3 36974 itg2addnclem 37320 itg2addnclem2 37321 pw2f1o2val2 42635 dnnumch3 42645 proot1mul 42796 proot1hash 42797 proot1ex 42798 wessf1ornlem 44729 preimafvsnel 46888 uniimaprimaeqfv 46891 elsetpreimafvbi 46900 |
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