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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 7234). (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 6068 | . 2 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) | |
2 | sqxpexg 7773 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V) | |
3 | ssexg 5328 | . 2 ⊢ ((( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → ( I ↾ 𝐴) ∈ V) | |
4 | 1, 2, 3 | sylancr 587 | 1 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 Vcvv 3477 ⊆ wss 3962 I cid 5581 × cxp 5686 ↾ cres 5690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-res 5700 |
This theorem is referenced by: ordiso 9553 wdomref 9609 dfac9 10174 relexp0g 15057 relexpsucnnr 15060 ndxarg 17229 idfu2nd 17927 idfu1st 17929 idfucl 17931 funcestrcsetclem4 18198 equivestrcsetc 18207 funcsetcestrclem4 18213 sursubmefmnd 18921 injsubmefmnd 18922 smndex1n0mnd 18937 islinds2 21850 pf1ind 22374 ausgrusgrb 29196 upgrres1lem1 29340 cusgrexilem1 29470 sizusglecusg 29495 pliguhgr 30514 bj-evalid 37058 bj-diagval 37156 poimirlem15 37621 xrnidresex 38388 dib0 41146 dicn0 41174 cdlemn11a 41189 dihord6apre 41238 dihatlat 41316 dihpN 41318 eldioph2lem1 42747 eldioph2lem2 42748 dfrtrcl5 43618 dfrcl2 43663 relexpiidm 43693 ushggricedg 47833 uspgrsprfo 47991 rngcidALTV 48117 ringcidALTV 48151 |
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