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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 5119 | . . . . . 6 ⊢ (A ∈ (𝐹‘B) ↔ ∃x(A ∈ x ∧ ∀y(B𝐹y ↔ y = x))) | |
2 | exsimpr 1506 | . . . . . 6 ⊢ (∃x(A ∈ x ∧ ∀y(B𝐹y ↔ y = x)) → ∃x∀y(B𝐹y ↔ y = x)) | |
3 | 1, 2 | sylbi 114 | . . . . 5 ⊢ (A ∈ (𝐹‘B) → ∃x∀y(B𝐹y ↔ y = x)) |
4 | equsb1 1665 | . . . . . . . 8 ⊢ [x / y]y = x | |
5 | spsbbi 1722 | . . . . . . . 8 ⊢ (∀y(B𝐹y ↔ y = x) → ([x / y]B𝐹y ↔ [x / y]y = x)) | |
6 | 4, 5 | mpbiri 157 | . . . . . . 7 ⊢ (∀y(B𝐹y ↔ y = x) → [x / y]B𝐹y) |
7 | nfv 1418 | . . . . . . . 8 ⊢ Ⅎy B𝐹x | |
8 | breq2 3759 | . . . . . . . 8 ⊢ (y = x → (B𝐹y ↔ B𝐹x)) | |
9 | 7, 8 | sbie 1671 | . . . . . . 7 ⊢ ([x / y]B𝐹y ↔ B𝐹x) |
10 | 6, 9 | sylib 127 | . . . . . 6 ⊢ (∀y(B𝐹y ↔ y = x) → B𝐹x) |
11 | 10 | eximi 1488 | . . . . 5 ⊢ (∃x∀y(B𝐹y ↔ y = x) → ∃x B𝐹x) |
12 | 3, 11 | syl 14 | . . . 4 ⊢ (A ∈ (𝐹‘B) → ∃x B𝐹x) |
13 | 12 | anim2i 324 | . . 3 ⊢ ((Rel 𝐹 ∧ A ∈ (𝐹‘B)) → (Rel 𝐹 ∧ ∃x B𝐹x)) |
14 | 19.42v 1783 | . . 3 ⊢ (∃x(Rel 𝐹 ∧ B𝐹x) ↔ (Rel 𝐹 ∧ ∃x B𝐹x)) | |
15 | 13, 14 | sylibr 137 | . 2 ⊢ ((Rel 𝐹 ∧ A ∈ (𝐹‘B)) → ∃x(Rel 𝐹 ∧ B𝐹x)) |
16 | releldm 4512 | . . 3 ⊢ ((Rel 𝐹 ∧ B𝐹x) → B ∈ dom 𝐹) | |
17 | 16 | exlimiv 1486 | . 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 97 ↔ wb 98 ∀wal 1240 ∃wex 1378 ∈ wcel 1390 [wsb 1642 class class class wbr 3755 dom cdm 4288 Rel wrel 4293 ‘cfv 4845 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-xp 4294 df-rel 4295 df-dm 4298 df-iota 4810 df-fv 4853 |
This theorem is referenced by: elmpt2cl 5640 mpt2xopn0yelv 5795 eluzel2 8254 |
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