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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 6245 | . 2 ⊢ (𝐹 Fn 𝐴 → (𝐶 ∈ (◡𝐹 “ {𝐵}) ↔ (𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) ∈ {𝐵}))) | |
2 | fvex 6113 | . . . 4 ⊢ (𝐹‘𝐶) ∈ V | |
3 | 2 | elsn 4140 | . . 3 ⊢ ((𝐹‘𝐶) ∈ {𝐵} ↔ (𝐹‘𝐶) = 𝐵) |
4 | 3 | anbi2i 726 | . 2 ⊢ ((𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) ∈ {𝐵}) ↔ (𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) = 𝐵)) |
5 | 1, 4 | syl6bb 275 | 1 ⊢ (𝐹 Fn 𝐴 → (𝐶 ∈ (◡𝐹 “ {𝐵}) ↔ (𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) = 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {csn 4125 ◡ccnv 5037 “ cima 5041 Fn wfn 5799 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 |
This theorem is referenced by: fparlem1 7164 fparlem2 7165 pw2f1olem 7949 recmulnq 9665 dmrecnq 9669 vdwlem1 15523 vdwlem2 15524 vdwlem6 15528 vdwlem8 15530 vdwlem9 15531 vdwlem12 15534 vdwlem13 15535 ramval 15550 ramub1lem1 15568 ghmeqker 17510 efgrelexlemb 17986 efgredeu 17988 psgnevpmb 19752 qtopeu 21329 itg1addlem1 23265 i1faddlem 23266 i1fmullem 23267 i1fmulclem 23275 i1fres 23278 itg10a 23283 itg1ge0a 23284 itg1climres 23287 mbfi1fseqlem4 23291 ply1remlem 23726 ply1rem 23727 fta1glem1 23729 fta1glem2 23730 fta1g 23731 fta1blem 23732 plyco0 23752 ofmulrt 23841 plyremlem 23863 plyrem 23864 fta1lem 23866 fta1 23867 vieta1lem1 23869 vieta1lem2 23870 vieta1 23871 plyexmo 23872 elaa 23875 aannenlem1 23887 aalioulem2 23892 pilem1 24009 efif1olem3 24094 efif1olem4 24095 efifo 24097 eff1olem 24098 basellem4 24610 lgsqrlem2 24872 lgsqrlem3 24873 rpvmasum2 25001 dirith 25018 foresf1o 28727 ofpreima 28848 1stpreimas 28866 locfinreflem 29235 qqhre 29392 indpi1 29411 indpreima 29414 sibfof 29729 cvmliftlem6 30526 cvmliftlem7 30527 cvmliftlem8 30528 cvmliftlem9 30529 taupilem3 32342 itg2addnclem 32631 itg2addnclem2 32632 pw2f1o2val2 36625 dnnumch3 36635 proot1mul 36796 proot1hash 36797 proot1ex 36798 wessf1ornlem 38366 |
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