Theorem fviss 6743

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 6705	. . . 4 ⊢ (𝑥 ∈ ( I ‘𝐴) → 𝐴 ∈ V)
3		fvi 6742	. . . 4 ⊢ (𝐴 ∈ V → ( I ‘𝐴) = 𝐴)
4	2, 3	syl 17	. . 3 ⊢ (𝑥 ∈ ( I ‘𝐴) → ( I ‘𝐴) = 𝐴)
5	1, 4	eleqtrd 2917	. 2 ⊢ (𝑥 ∈ ( I ‘𝐴) → 𝑥 ∈ 𝐴)
6	5	ssriv 3973	1 ⊢ ( I ‘𝐴) ⊆ 𝐴

Syntax hints:   = wceq 1537   ∈ wcel 2114  Vcvv 3496   ⊆ wss 3938   I cid 5461  ‘cfv 6357

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365

This theorem is referenced by:  efglem  18844  efgtf  18850  efgtlen  18854  efginvrel2  18855  efginvrel1  18856  efgsfo  18867  efgredlemg  18870  efgredleme  18871  efgredlemd  18872  efgredlemc  18873  efgredlem  18875  efgred  18876  efgcpbllemb  18883  frgpinv  18892  frgpuplem  18900  frgpupf  18901  frgpup1  18903