Theorem fviss 6894

Description: The value of the identity function is a subset of the argument. (An artifact of our function value definition.) (Contributed by Mario Carneiro, 27-Feb-2016.)

Assertion

Ref	Expression
fviss	⊢ ( I ‘𝐴) ⊆ 𝐴

Proof of Theorem fviss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝑥 ∈ ( I ‘𝐴) → 𝑥 ∈ ( I ‘𝐴))
2		elfvex 6852	. . . 4 ⊢ (𝑥 ∈ ( I ‘𝐴) → 𝐴 ∈ V)
3		fvi 6893	. . . 4 ⊢ (𝐴 ∈ V → ( I ‘𝐴) = 𝐴)
4	2, 3	syl 17	. . 3 ⊢ (𝑥 ∈ ( I ‘𝐴) → ( I ‘𝐴) = 𝐴)
5	1, 4	eleqtrd 2833	. 2 ⊢ (𝑥 ∈ ( I ‘𝐴) → 𝑥 ∈ 𝐴)
6	5	ssriv 3933	1 ⊢ ( I ‘𝐴) ⊆ 𝐴

Syntax hints:   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ⊆ wss 3897   I cid 5505  ‘cfv 6476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-iota 6432  df-fun 6478  df-fv 6484

This theorem is referenced by:  efglem  19623  efgtf  19629  efgtlen  19633  efginvrel2  19634  efginvrel1  19635  efgsfo  19646  efgredlemg  19649  efgredleme  19650  efgredlemd  19651  efgredlemc  19652  efgredlem  19654  efgred  19655  efgcpbllemb  19662  frgpinv  19671  frgpuplem  19679  frgpupf  19680  frgpup1  19682