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Mirrors > Home > MPE Home > Th. List > fnresi | Structured version Visualization version GIF version |
Description: The restricted identity relation is a function on the restricting class. (Contributed by NM, 27-Aug-2004.) (Proof shortened by BJ, 27-Dec-2023.) |
Ref | Expression |
---|---|
fnresi | ⊢ ( I ↾ 𝐴) Fn 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idfn 6630 | . 2 ⊢ I Fn V | |
2 | ssv 3969 | . 2 ⊢ 𝐴 ⊆ V | |
3 | fnssres 6625 | . 2 ⊢ (( I Fn V ∧ 𝐴 ⊆ V) → ( I ↾ 𝐴) Fn 𝐴) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ ( I ↾ 𝐴) Fn 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3444 ⊆ wss 3911 I cid 5531 ↾ cres 5636 Fn wfn 6492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-res 5646 df-fun 6499 df-fn 6500 |
This theorem is referenced by: f1oi 6823 fninfp 7121 fndifnfp 7123 fnnfpeq0 7125 fveqf1o 7250 weniso 7300 iordsmo 8304 fipreima 9305 dfac9 10077 smndex1n0mnd 18727 pmtrfinv 19248 ustuqtop3 23611 fta1blem 25549 qaa 25699 dfiop2 30737 symgcom2 31984 tocycfvres1 32008 tocycfvres2 32009 cvmliftlem4 33939 cvmliftlem5 33940 poimirlem15 36139 poimirlem22 36146 ltrnid 38644 dvsid 42699 dflinc2 46577 |
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