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Mirrors > Home > ILE Home > Th. List > fniniseg | 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 5307 | . 2 ⊢ (𝐹 Fn 𝐴 → (𝐶 ∈ (◡𝐹 “ {𝐵}) ↔ (𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) ∈ {𝐵}))) | |
2 | funfvex 5212 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐶 ∈ dom 𝐹) → (𝐹‘𝐶) ∈ V) | |
3 | elsng 3413 | . . . . 5 ⊢ ((𝐹‘𝐶) ∈ V → ((𝐹‘𝐶) ∈ {𝐵} ↔ (𝐹‘𝐶) = 𝐵)) | |
4 | 2, 3 | syl 14 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐶 ∈ dom 𝐹) → ((𝐹‘𝐶) ∈ {𝐵} ↔ (𝐹‘𝐶) = 𝐵)) |
5 | 4 | funfni 5019 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐶 ∈ 𝐴) → ((𝐹‘𝐶) ∈ {𝐵} ↔ (𝐹‘𝐶) = 𝐵)) |
6 | 5 | pm5.32da 439 | . 2 ⊢ (𝐹 Fn 𝐴 → ((𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) ∈ {𝐵}) ↔ (𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) = 𝐵))) |
7 | 1, 6 | bitrd 186 | 1 ⊢ (𝐹 Fn 𝐴 → (𝐶 ∈ (◡𝐹 “ {𝐵}) ↔ (𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) = 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1284 ∈ wcel 1433 Vcvv 2601 {csn 3398 ◡ccnv 4362 dom cdm 4363 “ cima 4366 Fun wfun 4916 Fn wfn 4917 ‘cfv 4922 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-fv 4930 |
This theorem is referenced by: (None) |
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