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Mirrors > Home > ILE Home > Th. List > fnsnfv | GIF version |
Description: Singleton of function value. (Contributed by NM, 22-May-1998.) |
Ref | Expression |
---|---|
fnsnfv | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2090 | . . . 4 ⊢ (𝑦 = (𝐹‘𝐵) ↔ (𝐹‘𝐵) = 𝑦) | |
2 | fnbrfvb 5345 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝑦 ↔ 𝐵𝐹𝑦)) | |
3 | 1, 2 | syl5bb 190 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝑦 = (𝐹‘𝐵) ↔ 𝐵𝐹𝑦)) |
4 | 3 | abbidv 2205 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {𝑦 ∣ 𝑦 = (𝐹‘𝐵)} = {𝑦 ∣ 𝐵𝐹𝑦}) |
5 | df-sn 3452 | . . 3 ⊢ {(𝐹‘𝐵)} = {𝑦 ∣ 𝑦 = (𝐹‘𝐵)} | |
6 | 5 | a1i 9 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = {𝑦 ∣ 𝑦 = (𝐹‘𝐵)}) |
7 | imasng 4797 | . . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐹 “ {𝐵}) = {𝑦 ∣ 𝐵𝐹𝑦}) | |
8 | 7 | adantl 271 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹 “ {𝐵}) = {𝑦 ∣ 𝐵𝐹𝑦}) |
9 | 4, 6, 8 | 3eqtr4d 2130 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵})) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1289 ∈ wcel 1438 {cab 2074 {csn 3446 class class class wbr 3845 “ cima 4441 Fn wfn 5010 ‘cfv 5015 |
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 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-sbc 2841 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-id 4120 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-fv 5023 |
This theorem is referenced by: fnimapr 5364 funfvdm 5367 fvco2 5373 fvimacnvi 5413 fsn2 5471 phplem4 6569 phplem4dom 6576 phplem4on 6581 fiintim 6637 fidcenumlemrks 6660 fidcenumlemr 6662 resunimafz0 10232 |
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