Theorem relelfvdm 5128

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 5099	. . . . . 6 ⊢ (A ∈ (𝐹‘B) ↔ ∃x(A ∈ x ∧ ∀y(B𝐹y ↔ y = x)))
2		exsimpr 1491	. . . . . 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 1402	. . . . . . . 8 ⊢ Ⅎy B𝐹x
8		breq2 3740	. . . . . . . 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 1473	. . . . 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 1768	. . 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 4494	. . 3 ⊢ ((Rel 𝐹 ∧ B𝐹x) → B ∈ dom 𝐹)
17	16	exlimiv 1471	. 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 1226  ∃wex 1362   ∈ wcel 1374  [wsb 1627   class class class wbr 3736  dom cdm 4270  Rel wrel 4275  ‘cfv 4827

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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3847  ax-pow 3899  ax-pr 3916

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  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 2897  df-in 2899  df-ss 2906  df-pw 3334  df-sn 3354  df-pr 3355  df-op 3357  df-uni 3553  df-br 3737  df-opab 3791  df-xp 4276  df-rel 4277  df-dm 4280  df-iota 4792  df-fv 4835

This theorem is referenced by:  elmpt2cl  5619  mpt2xopn0yelv  5774