Theorem fviss 6968

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 6929	. . . 4 ⊢ (𝑥 ∈ ( I ‘𝐴) → 𝐴 ∈ V)
3		fvi 6967	. . . 4 ⊢ (𝐴 ∈ V → ( I ‘𝐴) = 𝐴)
4	2, 3	syl 17	. . 3 ⊢ (𝑥 ∈ ( I ‘𝐴) → ( I ‘𝐴) = 𝐴)
5	1, 4	eleqtrd 2835	. 2 ⊢ (𝑥 ∈ ( I ‘𝐴) → 𝑥 ∈ 𝐴)
6	5	ssriv 3986	1 ⊢ ( I ‘𝐴) ⊆ 𝐴

Syntax hints:   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ⊆ wss 3948   I cid 5573  ‘cfv 6543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551

This theorem is referenced by:  efglem  19625  efgtf  19631  efgtlen  19635  efginvrel2  19636  efginvrel1  19637  efgsfo  19648  efgredlemg  19651  efgredleme  19652  efgredlemd  19653  efgredlemc  19654  efgredlem  19656  efgred  19657  efgcpbllemb  19664  frgpinv  19673  frgpuplem  19681  frgpupf  19682  frgpup1  19684