Theorem relrnfvex 5514

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 5512	. 2 ⊢ (Rel 𝐹 → (𝐹‘𝐴) ⊆ ∪ ran 𝐹)
2		uniexg 4424	. 2 ⊢ (ran 𝐹 ∈ V → ∪ ran 𝐹 ∈ V)
3		ssexg 4128	. 2 ⊢ (((𝐹‘𝐴) ⊆ ∪ ran 𝐹 ∧ ∪ ran 𝐹 ∈ V) → (𝐹‘𝐴) ∈ V)
4	1, 2, 3	syl2an 287	1 ⊢ ((Rel 𝐹 ∧ ran 𝐹 ∈ V) → (𝐹‘𝐴) ∈ V)

Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 2141  Vcvv 2730   ⊆ wss 3121  ∪ cuni 3796  ran crn 4612  Rel wrel 4616  ‘cfv 5198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622  df-iota 5160  df-fv 5206

This theorem is referenced by: (None)