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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 6945 | . . . 4 ⊢ (𝑥 ∈ ( I ‘𝐴) → 𝐴 ∈ V) | |
3 | fvi 6985 | . . . 4 ⊢ (𝐴 ∈ V → ( I ‘𝐴) = 𝐴) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝑥 ∈ ( I ‘𝐴) → ( I ‘𝐴) = 𝐴) |
5 | 1, 4 | eleqtrd 2841 | . 2 ⊢ (𝑥 ∈ ( I ‘𝐴) → 𝑥 ∈ 𝐴) |
6 | 5 | ssriv 3999 | 1 ⊢ ( I ‘𝐴) ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 I cid 5582 ‘cfv 6563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 |
This theorem is referenced by: efglem 19749 efgtf 19755 efgtlen 19759 efginvrel2 19760 efginvrel1 19761 efgsfo 19772 efgredlemg 19775 efgredleme 19776 efgredlemd 19777 efgredlemc 19778 efgredlem 19780 efgred 19781 efgcpbllemb 19788 frgpinv 19797 frgpuplem 19805 frgpupf 19806 frgpup1 19808 |
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