Theorem relrnfvex 5612

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 5610	. 2 ⊢ (Rel 𝐹 → (𝐹‘𝐴) ⊆ ∪ ran 𝐹)
2		uniexg 4499	. 2 ⊢ (ran 𝐹 ∈ V → ∪ ran 𝐹 ∈ V)
3		ssexg 4194	. 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 2177  Vcvv 2773   ⊆ wss 3170  ∪ cuni 3859  ran crn 4689  Rel wrel 4693  ‘cfv 5285

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-br 4055  df-opab 4117  df-xp 4694  df-rel 4695  df-cnv 4696  df-dm 4698  df-rn 4699  df-iota 5246  df-fv 5293

This theorem is referenced by: (None)