Theorem relfvssunirn 5570

Description: The result of a function value is always a subset of the union of the range, even if it is invalid and thus empty. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 24-May-2019.)

Assertion

Ref	Expression
relfvssunirn	⊢ (Rel 𝐹 → (𝐹‘𝐴) ⊆ ∪ ran 𝐹)

Proof of Theorem relfvssunirn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relelrn 4898	. . . . 5 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝑥) → 𝑥 ∈ ran 𝐹)
2	1	ex 115	. . . 4 ⊢ (Rel 𝐹 → (𝐴𝐹𝑥 → 𝑥 ∈ ran 𝐹))
3		elssuni 3863	. . . 4 ⊢ (𝑥 ∈ ran 𝐹 → 𝑥 ⊆ ∪ ran 𝐹)
4	2, 3	syl6 33	. . 3 ⊢ (Rel 𝐹 → (𝐴𝐹𝑥 → 𝑥 ⊆ ∪ ran 𝐹))
5	4	alrimiv 1885	. 2 ⊢ (Rel 𝐹 → ∀𝑥(𝐴𝐹𝑥 → 𝑥 ⊆ ∪ ran 𝐹))
6		fvss 5568	. 2 ⊢ (∀𝑥(𝐴𝐹𝑥 → 𝑥 ⊆ ∪ ran 𝐹) → (𝐹‘𝐴) ⊆ ∪ ran 𝐹)
7	5, 6	syl 14	1 ⊢ (Rel 𝐹 → (𝐹‘𝐴) ⊆ ∪ ran 𝐹)

Syntax hints:   → wi 4  ∀wal 1362   ∈ wcel 2164   ⊆ wss 3153  ∪ cuni 3835   class class class wbr 4029  ran crn 4660  Rel wrel 4664  ‘cfv 5254

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667  df-dm 4669  df-rn 4670  df-iota 5215  df-fv 5262

This theorem is referenced by:  relrnfvex  5572