Theorem fvssunirng 5614

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 2779	. . . . 5 ⊢ 𝑥 ∈ V
2		brelrng 4928	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V ∧ 𝐴𝐹𝑥) → 𝑥 ∈ ran 𝐹)
3	2	3exp 1205	. . . . 5 ⊢ (𝐴 ∈ V → (𝑥 ∈ V → (𝐴𝐹𝑥 → 𝑥 ∈ ran 𝐹)))
4	1, 3	mpi 15	. . . 4 ⊢ (𝐴 ∈ V → (𝐴𝐹𝑥 → 𝑥 ∈ ran 𝐹))
5		elssuni 3892	. . . 4 ⊢ (𝑥 ∈ ran 𝐹 → 𝑥 ⊆ ∪ ran 𝐹)
6	4, 5	syl6 33	. . 3 ⊢ (𝐴 ∈ V → (𝐴𝐹𝑥 → 𝑥 ⊆ ∪ ran 𝐹))
7	6	alrimiv 1898	. 2 ⊢ (𝐴 ∈ V → ∀𝑥(𝐴𝐹𝑥 → 𝑥 ⊆ ∪ ran 𝐹))
8		fvss 5613	. 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 1371   ∈ wcel 2178  Vcvv 2776   ⊆ wss 3174  ∪ cuni 3864   class class class wbr 4059  ran crn 4694  ‘cfv 5290

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-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

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 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-cnv 4701  df-dm 4703  df-rn 4704  df-iota 5251  df-fv 5298

This theorem is referenced by:  fvexg  5618  ovssunirng  6002  strfvssn  12969  ptex  13211  prdsvallem  13219  prdsval  13220  xmetunirn  14945  mopnval  15029