Theorem relfvssunirn 5530

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 4862	. . . . 5 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝑥) → 𝑥 ∈ ran 𝐹)
2	1	ex 115	. . . 4 ⊢ (Rel 𝐹 → (𝐴𝐹𝑥 → 𝑥 ∈ ran 𝐹))
3		elssuni 3837	. . . 4 ⊢ (𝑥 ∈ ran 𝐹 → 𝑥 ⊆ ∪ ran 𝐹)
4	2, 3	syl6 33	. . 3 ⊢ (Rel 𝐹 → (𝐴𝐹𝑥 → 𝑥 ⊆ ∪ ran 𝐹))
5	4	alrimiv 1874	. 2 ⊢ (Rel 𝐹 → ∀𝑥(𝐴𝐹𝑥 → 𝑥 ⊆ ∪ ran 𝐹))
6		fvss 5528	. 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 1351   ∈ wcel 2148   ⊆ wss 3129  ∪ cuni 3809   class class class wbr 4002  ran crn 4626  Rel wrel 4630  ‘cfv 5215

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-xp 4631  df-rel 4632  df-cnv 4633  df-dm 4635  df-rn 4636  df-iota 5177  df-fv 5223

This theorem is referenced by:  relrnfvex  5532