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| Mirrors > Home > MPE Home > Th. List > fviss | Structured version Visualization version GIF version | ||
| 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.) | 
| Ref | Expression | 
|---|---|
| fviss | ⊢ ( I ‘𝐴) ⊆ 𝐴 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | id 22 | . . 3 ⊢ (𝑥 ∈ ( I ‘𝐴) → 𝑥 ∈ ( I ‘𝐴)) | |
| 2 | elfvex 6943 | . . . 4 ⊢ (𝑥 ∈ ( I ‘𝐴) → 𝐴 ∈ V) | |
| 3 | fvi 6984 | . . . 4 ⊢ (𝐴 ∈ V → ( I ‘𝐴) = 𝐴) | |
| 4 | 2, 3 | syl 17 | . . 3 ⊢ (𝑥 ∈ ( I ‘𝐴) → ( I ‘𝐴) = 𝐴) | 
| 5 | 1, 4 | eleqtrd 2842 | . 2 ⊢ (𝑥 ∈ ( I ‘𝐴) → 𝑥 ∈ 𝐴) | 
| 6 | 5 | ssriv 3986 | 1 ⊢ ( I ‘𝐴) ⊆ 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∈ wcel 2107 Vcvv 3479 ⊆ wss 3950 I cid 5576 ‘cfv 6560 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 | 
| This theorem is referenced by: efglem 19735 efgtf 19741 efgtlen 19745 efginvrel2 19746 efginvrel1 19747 efgsfo 19758 efgredlemg 19761 efgredleme 19762 efgredlemd 19763 efgredlemc 19764 efgredlem 19766 efgred 19767 efgcpbllemb 19774 frgpinv 19783 frgpuplem 19791 frgpupf 19792 frgpup1 19794 | 
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