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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. (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 6131 | . . . 4 ⊢ (𝑥 ∈ ( I ‘𝐴) → 𝐴 ∈ V) | |
3 | fvi 6165 | . . . 4 ⊢ (𝐴 ∈ V → ( I ‘𝐴) = 𝐴) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝑥 ∈ ( I ‘𝐴) → ( I ‘𝐴) = 𝐴) |
5 | 1, 4 | eleqtrd 2690 | . 2 ⊢ (𝑥 ∈ ( I ‘𝐴) → 𝑥 ∈ 𝐴) |
6 | 5 | ssriv 3572 | 1 ⊢ ( I ‘𝐴) ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 I cid 4948 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 |
This theorem is referenced by: efglem 17952 efgtf 17958 efgtlen 17962 efginvrel2 17963 efginvrel1 17964 efgsfo 17975 efgredlemg 17978 efgredleme 17979 efgredlemd 17980 efgredlemc 17981 efgredlem 17983 efgred 17984 efgcpbllemb 17991 frgpinv 18000 frgpuplem 18008 frgpupf 18009 frgpup1 18011 |
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