Theorem relrnfvex 5647

Description: If a function has a set range, then the function value exists unconditional on the domain. (Contributed by Mario Carneiro, 24-May-2019.)

Assertion

Ref	Expression
relrnfvex	⊢ ((Rel 𝐹 ∧ ran 𝐹 ∈ V) → (𝐹‘𝐴) ∈ V)

Proof of Theorem relrnfvex

Step	Hyp	Ref	Expression
1		relfvssunirn 5645	. 2 ⊢ (Rel 𝐹 → (𝐹‘𝐴) ⊆ ∪ ran 𝐹)
2		uniexg 4530	. 2 ⊢ (ran 𝐹 ∈ V → ∪ ran 𝐹 ∈ V)
3		ssexg 4223	. 2 ⊢ (((𝐹‘𝐴) ⊆ ∪ ran 𝐹 ∧ ∪ ran 𝐹 ∈ V) → (𝐹‘𝐴) ∈ V)
4	1, 2, 3	syl2an 289	1 ⊢ ((Rel 𝐹 ∧ ran 𝐹 ∈ V) → (𝐹‘𝐴) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2200  Vcvv 2799   ⊆ wss 3197  ∪ cuni 3888  ran crn 4720  Rel wrel 4724  ‘cfv 5318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-cnv 4727  df-dm 4729  df-rn 4730  df-iota 5278  df-fv 5326

This theorem is referenced by: (None)