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Mirrors > Home > MPE Home > Th. List > resiexg | Structured version Visualization version GIF version |
Description: The existence of a restricted identity function, proved without using the Axiom of Replacement (unlike resfunexg 7217). (Contributed by NM, 13-Jan-2007.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
Ref | Expression |
---|---|
resiexg | ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idssxp 6049 | . 2 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) | |
2 | sqxpexg 7742 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V) | |
3 | ssexg 5324 | . 2 ⊢ ((( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → ( I ↾ 𝐴) ∈ V) | |
4 | 1, 2, 3 | sylancr 588 | 1 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3475 ⊆ wss 3949 I cid 5574 × cxp 5675 ↾ cres 5679 |
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 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
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 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-res 5689 |
This theorem is referenced by: ordiso 9511 wdomref 9567 dfac9 10131 relexp0g 14969 relexpsucnnr 14972 ndxarg 17129 idfu2nd 17827 idfu1st 17829 idfucl 17831 funcestrcsetclem4 18095 equivestrcsetc 18104 funcsetcestrclem4 18110 sursubmefmnd 18777 injsubmefmnd 18778 smndex1n0mnd 18793 islinds2 21368 pf1ind 21874 ausgrusgrb 28425 upgrres1lem1 28566 cusgrexilem1 28696 sizusglecusg 28720 pliguhgr 29739 bj-evalid 35957 bj-diagval 36055 poimirlem15 36503 xrnidresex 37277 dib0 40035 dicn0 40063 cdlemn11a 40078 dihord6apre 40127 dihatlat 40205 dihpN 40207 eldioph2lem1 41498 eldioph2lem2 41499 dfrtrcl5 42380 dfrcl2 42425 relexpiidm 42455 uspgrsprfo 46526 rngcidALTV 46889 ringcidALTV 46952 |
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