Theorem fviss 6740

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 6702	. . . 4 ⊢ (𝑥 ∈ ( I ‘𝐴) → 𝐴 ∈ V)
3		fvi 6739	. . . 4 ⊢ (𝐴 ∈ V → ( I ‘𝐴) = 𝐴)
4	2, 3	syl 17	. . 3 ⊢ (𝑥 ∈ ( I ‘𝐴) → ( I ‘𝐴) = 𝐴)
5	1, 4	eleqtrd 2915	. 2 ⊢ (𝑥 ∈ ( I ‘𝐴) → 𝑥 ∈ 𝐴)
6	5	ssriv 3970	1 ⊢ ( I ‘𝐴) ⊆ 𝐴

Syntax hints:   = wceq 1533   ∈ wcel 2110  Vcvv 3494   ⊆ wss 3935   I cid 5458  ‘cfv 6354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fun 6356  df-fv 6362

This theorem is referenced by:  efglem  18841  efgtf  18847  efgtlen  18851  efginvrel2  18852  efginvrel1  18853  efgsfo  18864  efgredlemg  18867  efgredleme  18868  efgredlemd  18869  efgredlemc  18870  efgredlem  18872  efgred  18873  efgcpbllemb  18880  frgpinv  18889  frgpuplem  18897  frgpupf  18898  frgpup1  18900