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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 6678 | . . . 4 ⊢ (𝑥 ∈ ( I ‘𝐴) → 𝐴 ∈ V) | |
3 | fvi 6715 | . . . 4 ⊢ (𝐴 ∈ V → ( I ‘𝐴) = 𝐴) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝑥 ∈ ( I ‘𝐴) → ( I ‘𝐴) = 𝐴) |
5 | 1, 4 | eleqtrd 2892 | . 2 ⊢ (𝑥 ∈ ( I ‘𝐴) → 𝑥 ∈ 𝐴) |
6 | 5 | ssriv 3919 | 1 ⊢ ( I ‘𝐴) ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 I cid 5424 ‘cfv 6324 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 |
This theorem is referenced by: efglem 18834 efgtf 18840 efgtlen 18844 efginvrel2 18845 efginvrel1 18846 efgsfo 18857 efgredlemg 18860 efgredleme 18861 efgredlemd 18862 efgredlemc 18863 efgredlem 18865 efgred 18866 efgcpbllemb 18873 frgpinv 18882 frgpuplem 18890 frgpupf 18891 frgpup1 18893 |
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