Theorem relelfvdm 5097

Description: If a function value has a member, the argument belongs to the domain. (Contributed by Jim Kingdon, 22-Jan-2019.)

Assertion

Ref	Expression
relelfvdm	⊢ ((Rel 𝐹 ∧ A ∈ (𝐹‘B)) → B ∈ dom 𝐹)

Proof of Theorem relelfvdm

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfv 5068	. . . . . 6 ⊢ (A ∈ (𝐹‘B) ↔ ∃x(A ∈ x ∧ ∀y(B𝐹y ↔ y = x)))
2		exsimpr 1492	. . . . . 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 1650	. . . . . . . 8 ⊢ [x / y]y = x
5		spsbbi 1707	. . . . . . . 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 1403	. . . . . . . 8 ⊢ Ⅎy B𝐹x
8		breq2 3720	. . . . . . . 8 ⊢ (y = x → (B𝐹y ↔ B𝐹x))
9	7, 8	sbie 1656	. . . . . . 7 ⊢ ([x / y]B𝐹y ↔ B𝐹x)
10	6, 9	sylib 127	. . . . . 6 ⊢ (∀y(B𝐹y ↔ y = x) → B𝐹x)
11	10	eximi 1474	. . . . 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 1769	. . 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 4462	. . 3 ⊢ ((Rel 𝐹 ∧ B𝐹x) → B ∈ dom 𝐹)
17	16	exlimiv 1472	. 2 ⊢ (∃x(Rel 𝐹 ∧ B𝐹x) → B ∈ dom 𝐹)
18	15, 17	syl 14	1 ⊢ ((Rel 𝐹 ∧ A ∈ (𝐹‘B)) → B ∈ dom 𝐹)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1314  ∃wex 1361   ∈ wcel 1375  [wsb 1627   class class class wbr 3716  dom cdm 4238  Rel wrel 4243  ‘cfv 4796

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 617  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-sep 3827  ax-pow 3879  ax-pr 3896

This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-un 2900  df-in 2902  df-ss 2909  df-pw 3313  df-sn 3333  df-pr 3334  df-op 3336  df-uni 3533  df-br 3717  df-opab 3771  df-xp 4244  df-rel 4245  df-dm 4248  df-iota 4761  df-fv 4804

This theorem is referenced by:  elmpt2cl  5587  mpt2xopn0yelv  5742