Theorem relfvssunirn 5523

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 4856	. . . . 5 ⊢ ((Rel 𝐹 ∧ 𝐴𝐹𝑥) → 𝑥 ∈ ran 𝐹)
2	1	ex 115	. . . 4 ⊢ (Rel 𝐹 → (𝐴𝐹𝑥 → 𝑥 ∈ ran 𝐹))
3		elssuni 3833	. . . 4 ⊢ (𝑥 ∈ ran 𝐹 → 𝑥 ⊆ ∪ ran 𝐹)
4	2, 3	syl6 33	. . 3 ⊢ (Rel 𝐹 → (𝐴𝐹𝑥 → 𝑥 ⊆ ∪ ran 𝐹))
5	4	alrimiv 1872	. 2 ⊢ (Rel 𝐹 → ∀𝑥(𝐴𝐹𝑥 → 𝑥 ⊆ ∪ ran 𝐹))
6		fvss 5521	. 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 2146   ⊆ wss 3127  ∪ cuni 3805   class class class wbr 3998  ran crn 4621  Rel wrel 4625  ‘cfv 5208

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-xp 4626  df-rel 4627  df-cnv 4628  df-dm 4630  df-rn 4631  df-iota 5170  df-fv 5216

This theorem is referenced by:  relrnfvex  5525