Theorem relelfvdm 5148

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 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)

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-bnd 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  8214