Theorem fvssunirng 5576

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 2766	. . . . 5 ⊢ 𝑥 ∈ V
2		brelrng 4898	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V ∧ 𝐴𝐹𝑥) → 𝑥 ∈ ran 𝐹)
3	2	3exp 1204	. . . . 5 ⊢ (𝐴 ∈ V → (𝑥 ∈ V → (𝐴𝐹𝑥 → 𝑥 ∈ ran 𝐹)))
4	1, 3	mpi 15	. . . 4 ⊢ (𝐴 ∈ V → (𝐴𝐹𝑥 → 𝑥 ∈ ran 𝐹))
5		elssuni 3868	. . . 4 ⊢ (𝑥 ∈ ran 𝐹 → 𝑥 ⊆ ∪ ran 𝐹)
6	4, 5	syl6 33	. . 3 ⊢ (𝐴 ∈ V → (𝐴𝐹𝑥 → 𝑥 ⊆ ∪ ran 𝐹))
7	6	alrimiv 1888	. 2 ⊢ (𝐴 ∈ V → ∀𝑥(𝐴𝐹𝑥 → 𝑥 ⊆ ∪ ran 𝐹))
8		fvss 5575	. 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 1362   ∈ wcel 2167  Vcvv 2763   ⊆ wss 3157  ∪ cuni 3840   class class class wbr 4034  ran crn 4665  ‘cfv 5259

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-cnv 4672  df-dm 4674  df-rn 4675  df-iota 5220  df-fv 5267

This theorem is referenced by:  fvexg  5580  ovssunirng  5960  strfvssn  12725  ptex  12966  prdsvallem  12974  prdsval  12975  xmetunirn  14678  mopnval  14762