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Mirrors > Home > ILE Home > Th. List > relelfvdm | GIF version |
Description: If a function value has a member, the argument belongs to the domain. (Contributed by Jim Kingdon, 22-Jan-2019.) |
Ref | Expression |
---|---|
relelfvdm | ⊢ ((Rel 𝐹 ∧ 𝐴 ∈ (𝐹‘𝐵)) → 𝐵 ∈ dom 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfv 5419 | . . . . . 6 ⊢ (𝐴 ∈ (𝐹‘𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥))) | |
2 | exsimpr 1597 | . . . . . 6 ⊢ (∃𝑥(𝐴 ∈ 𝑥 ∧ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)) → ∃𝑥∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)) | |
3 | 1, 2 | sylbi 120 | . . . . 5 ⊢ (𝐴 ∈ (𝐹‘𝐵) → ∃𝑥∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)) |
4 | equsb1 1758 | . . . . . . . 8 ⊢ [𝑥 / 𝑦]𝑦 = 𝑥 | |
5 | spsbbi 1816 | . . . . . . . 8 ⊢ (∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥) → ([𝑥 / 𝑦]𝐵𝐹𝑦 ↔ [𝑥 / 𝑦]𝑦 = 𝑥)) | |
6 | 4, 5 | mpbiri 167 | . . . . . . 7 ⊢ (∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥) → [𝑥 / 𝑦]𝐵𝐹𝑦) |
7 | nfv 1508 | . . . . . . . 8 ⊢ Ⅎ𝑦 𝐵𝐹𝑥 | |
8 | breq2 3933 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝐵𝐹𝑦 ↔ 𝐵𝐹𝑥)) | |
9 | 7, 8 | sbie 1764 | . . . . . . 7 ⊢ ([𝑥 / 𝑦]𝐵𝐹𝑦 ↔ 𝐵𝐹𝑥) |
10 | 6, 9 | sylib 121 | . . . . . 6 ⊢ (∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥) → 𝐵𝐹𝑥) |
11 | 10 | eximi 1579 | . . . . 5 ⊢ (∃𝑥∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥) → ∃𝑥 𝐵𝐹𝑥) |
12 | 3, 11 | syl 14 | . . . 4 ⊢ (𝐴 ∈ (𝐹‘𝐵) → ∃𝑥 𝐵𝐹𝑥) |
13 | 12 | anim2i 339 | . . 3 ⊢ ((Rel 𝐹 ∧ 𝐴 ∈ (𝐹‘𝐵)) → (Rel 𝐹 ∧ ∃𝑥 𝐵𝐹𝑥)) |
14 | 19.42v 1878 | . . 3 ⊢ (∃𝑥(Rel 𝐹 ∧ 𝐵𝐹𝑥) ↔ (Rel 𝐹 ∧ ∃𝑥 𝐵𝐹𝑥)) | |
15 | 13, 14 | sylibr 133 | . 2 ⊢ ((Rel 𝐹 ∧ 𝐴 ∈ (𝐹‘𝐵)) → ∃𝑥(Rel 𝐹 ∧ 𝐵𝐹𝑥)) |
16 | releldm 4774 | . . 3 ⊢ ((Rel 𝐹 ∧ 𝐵𝐹𝑥) → 𝐵 ∈ dom 𝐹) | |
17 | 16 | exlimiv 1577 | . 2 ⊢ (∃𝑥(Rel 𝐹 ∧ 𝐵𝐹𝑥) → 𝐵 ∈ dom 𝐹) |
18 | 15, 17 | syl 14 | 1 ⊢ ((Rel 𝐹 ∧ 𝐴 ∈ (𝐹‘𝐵)) → 𝐵 ∈ dom 𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1329 ∃wex 1468 ∈ wcel 1480 [wsb 1735 class class class wbr 3929 dom cdm 4539 Rel wrel 4544 ‘cfv 5123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-dm 4549 df-iota 5088 df-fv 5131 |
This theorem is referenced by: mptrcl 5503 elfvmptrab1 5515 elmpocl 5968 oprssdmm 6069 mpoxopn0yelv 6136 eluzel2 9331 hashinfom 10524 istopon 12180 istps 12199 topontopn 12204 eltg4i 12224 eltg3 12226 tg1 12228 tg2 12229 tgclb 12234 cldrcl 12271 neiss2 12311 lmrcl 12360 cnprcl2k 12375 metflem 12518 xmetf 12519 ismet2 12523 xmeteq0 12528 xmettri2 12530 xmetpsmet 12538 xmetres2 12548 blfvalps 12554 blex 12556 blvalps 12557 blval 12558 blfps 12578 blf 12579 mopnval 12611 isxms2 12621 comet 12668 |
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