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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 5427 | . . . . . 6 ⊢ (𝐴 ∈ (𝐹‘𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥))) | |
2 | exsimpr 1598 | . . . . . 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 1759 | . . . . . . . 8 ⊢ [𝑥 / 𝑦]𝑦 = 𝑥 | |
5 | spsbbi 1817 | . . . . . . . 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 1509 | . . . . . . . 8 ⊢ Ⅎ𝑦 𝐵𝐹𝑥 | |
8 | breq2 3941 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝐵𝐹𝑦 ↔ 𝐵𝐹𝑥)) | |
9 | 7, 8 | sbie 1765 | . . . . . . 7 ⊢ ([𝑥 / 𝑦]𝐵𝐹𝑦 ↔ 𝐵𝐹𝑥) |
10 | 6, 9 | sylib 121 | . . . . . 6 ⊢ (∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥) → 𝐵𝐹𝑥) |
11 | 10 | eximi 1580 | . . . . 5 ⊢ (∃𝑥∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥) → ∃𝑥 𝐵𝐹𝑥) |
12 | 3, 11 | syl 14 | . . . 4 ⊢ (𝐴 ∈ (𝐹‘𝐵) → ∃𝑥 𝐵𝐹𝑥) |
13 | 12 | anim2i 340 | . . 3 ⊢ ((Rel 𝐹 ∧ 𝐴 ∈ (𝐹‘𝐵)) → (Rel 𝐹 ∧ ∃𝑥 𝐵𝐹𝑥)) |
14 | 19.42v 1879 | . . 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 4782 | . . 3 ⊢ ((Rel 𝐹 ∧ 𝐵𝐹𝑥) → 𝐵 ∈ dom 𝐹) | |
17 | 16 | exlimiv 1578 | . 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 1330 ∃wex 1469 ∈ wcel 1481 [wsb 1736 class class class wbr 3937 dom cdm 4547 Rel wrel 4552 ‘cfv 5131 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-rel 4554 df-dm 4557 df-iota 5096 df-fv 5139 |
This theorem is referenced by: mptrcl 5511 elfvmptrab1 5523 elmpocl 5976 oprssdmm 6077 mpoxopn0yelv 6144 eluzel2 9355 hashinfom 10556 istopon 12219 istps 12238 topontopn 12243 eltg4i 12263 eltg3 12265 tg1 12267 tg2 12268 tgclb 12273 cldrcl 12310 neiss2 12350 lmrcl 12399 cnprcl2k 12414 metflem 12557 xmetf 12558 ismet2 12562 xmeteq0 12567 xmettri2 12569 xmetpsmet 12577 xmetres2 12587 blfvalps 12593 blex 12595 blvalps 12596 blval 12597 blfps 12617 blf 12618 mopnval 12650 isxms2 12660 comet 12707 |
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