Theorem fvssunirng 5690

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

Assertion

Ref	Expression
fvssunirng	⊢ (𝐴 ∈ V → (𝐹‘𝐴) ⊆ ∪ ran 𝐹)

Proof of Theorem fvssunirng

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2818	. . . . 5 ⊢ 𝑥 ∈ V
2		brelrng 4993	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V ∧ 𝐴𝐹𝑥) → 𝑥 ∈ ran 𝐹)
3	2	3exp 1229	. . . . 5 ⊢ (𝐴 ∈ V → (𝑥 ∈ V → (𝐴𝐹𝑥 → 𝑥 ∈ ran 𝐹)))
4	1, 3	mpi 15	. . . 4 ⊢ (𝐴 ∈ V → (𝐴𝐹𝑥 → 𝑥 ∈ ran 𝐹))
5		elssuni 3947	. . . 4 ⊢ (𝑥 ∈ ran 𝐹 → 𝑥 ⊆ ∪ ran 𝐹)
6	4, 5	syl6 33	. . 3 ⊢ (𝐴 ∈ V → (𝐴𝐹𝑥 → 𝑥 ⊆ ∪ ran 𝐹))
7	6	alrimiv 1923	. 2 ⊢ (𝐴 ∈ V → ∀𝑥(𝐴𝐹𝑥 → 𝑥 ⊆ ∪ ran 𝐹))
8		fvss 5689	. 2 ⊢ (∀𝑥(𝐴𝐹𝑥 → 𝑥 ⊆ ∪ ran 𝐹) → (𝐹‘𝐴) ⊆ ∪ ran 𝐹)
9	7, 8	syl 14	1 ⊢ (𝐴 ∈ V → (𝐹‘𝐴) ⊆ ∪ ran 𝐹)

Syntax hints:   → wi 4  ∀wal 1396   ∈ wcel 2205  Vcvv 2815   ⊆ wss 3214  ∪ cuni 3919   class class class wbr 4114  ran crn 4755  ‘cfv 5357

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-cnv 4762  df-dm 4764  df-rn 4765  df-iota 5317  df-fv 5365

This theorem is referenced by:  fvexg  5694  ovssunirng  6093  strfvssn  13318  ptex  13561  prdsvallem  13564  prdsval  14115  xmetunirn  15349  mopnval  15433