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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 7078 | . 2 ⊢ (𝐹 Fn 𝐴 → (𝐶 ∈ (◡𝐹 “ {𝐵}) ↔ (𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) ∈ {𝐵}))) | |
| 2 | fvex 6919 | . . . 4 ⊢ (𝐹‘𝐶) ∈ V | |
| 3 | 2 | elsn 4641 | . . 3 ⊢ ((𝐹‘𝐶) ∈ {𝐵} ↔ (𝐹‘𝐶) = 𝐵) |
| 4 | 3 | anbi2i 623 | . 2 ⊢ ((𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) ∈ {𝐵}) ↔ (𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) = 𝐵)) |
| 5 | 1, 4 | bitrdi 287 | 1 ⊢ (𝐹 Fn 𝐴 → (𝐶 ∈ (◡𝐹 “ {𝐵}) ↔ (𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) = 𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {csn 4626 ◡ccnv 5684 “ cima 5688 Fn wfn 6556 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 |
| This theorem is referenced by: fparlem1 8137 fparlem2 8138 pw2f1olem 9116 recmulnq 11004 dmrecnq 11008 vdwlem1 17019 vdwlem2 17020 vdwlem6 17024 vdwlem8 17026 vdwlem9 17027 vdwlem12 17030 vdwlem13 17031 ramval 17046 ramub1lem1 17064 ghmeqker 19261 ghmqusnsglem1 19298 ghmquskerlem1 19301 ghmqusker 19305 efgrelexlemb 19768 efgredeu 19770 psgnevpmb 21605 qtopeu 23724 itg1addlem1 25727 i1faddlem 25728 i1fmullem 25729 i1fmulclem 25737 i1fres 25740 itg10a 25745 itg1ge0a 25746 itg1climres 25749 mbfi1fseqlem4 25753 ply1remlem 26204 ply1rem 26205 fta1glem1 26207 fta1glem2 26208 fta1g 26209 fta1blem 26210 plyco0 26231 ofmulrt 26323 plyremlem 26346 plyrem 26347 fta1lem 26349 fta1 26350 vieta1lem1 26352 vieta1lem2 26353 vieta1 26354 plyexmo 26355 elaa 26358 aannenlem1 26370 aalioulem2 26375 pilem1 26495 efif1olem3 26586 efif1olem4 26587 efifo 26589 eff1olem 26590 basellem4 27127 lgsqrlem2 27391 lgsqrlem3 27392 rpvmasum2 27556 dirith 27573 foresf1o 32523 ofpreima 32675 fnpreimac 32681 1stpreimas 32715 indpi1 32845 indpreima 32850 s3clhash 32932 pwrssmgc 32990 cycpmconjslem2 33175 cyc3conja 33177 exsslsb 33647 dimkerim 33678 elirng 33736 irngss 33737 irngnzply1 33741 locfinreflem 33839 qqhre 34021 sibfof 34342 cvmliftlem6 35295 cvmliftlem7 35296 cvmliftlem8 35297 cvmliftlem9 35298 taupilem3 37320 itg2addnclem 37678 itg2addnclem2 37679 pw2f1o2val2 43052 dnnumch3 43059 proot1mul 43206 proot1hash 43207 proot1ex 43208 wessf1ornlem 45190 preimafvsnel 47366 uniimaprimaeqfv 47369 elsetpreimafvbi 47378 |
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