Theorem fviss 6939

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 6897	. . . 4 ⊢ (𝑥 ∈ ( I ‘𝐴) → 𝐴 ∈ V)
3		fvi 6938	. . . 4 ⊢ (𝐴 ∈ V → ( I ‘𝐴) = 𝐴)
4	2, 3	syl 17	. . 3 ⊢ (𝑥 ∈ ( I ‘𝐴) → ( I ‘𝐴) = 𝐴)
5	1, 4	eleqtrd 2863	. 2 ⊢ (𝑥 ∈ ( I ‘𝐴) → 𝑥 ∈ 𝐴)
6	5	ssriv 3938	1 ⊢ ( I ‘𝐴) ⊆ 𝐴

Syntax hints:   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ⊆ wss 3902   I cid 5537  ‘cfv 6516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-iota 6472  df-fun 6518  df-fv 6524

This theorem is referenced by:  efglem  19747  efgtf  19753  efgtlen  19757  efginvrel2  19758  efginvrel1  19759  efgsfo  19770  efgredlemg  19773  efgredleme  19774  efgredlemd  19775  efgredlemc  19776  efgredlem  19778  efgred  19779  efgcpbllemb  19786  frgpinv  19795  frgpuplem  19803  frgpupf  19804  frgpup1  19806