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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 7235). (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 6067 | . 2 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) | |
| 2 | sqxpexg 7775 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V) | |
| 3 | ssexg 5323 | . 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 2108 Vcvv 3480 ⊆ wss 3951 I cid 5577 × cxp 5683 ↾ cres 5687 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-res 5697 |
| This theorem is referenced by: ordiso 9556 wdomref 9612 dfac9 10177 relexp0g 15061 relexpsucnnr 15064 ndxarg 17233 idfu2nd 17922 idfu1st 17924 idfucl 17926 funcestrcsetclem4 18188 equivestrcsetc 18197 funcsetcestrclem4 18203 sursubmefmnd 18909 injsubmefmnd 18910 smndex1n0mnd 18925 islinds2 21833 pf1ind 22359 ausgrusgrb 29182 upgrres1lem1 29326 cusgrexilem1 29456 sizusglecusg 29481 pliguhgr 30505 bj-evalid 37077 bj-diagval 37175 poimirlem15 37642 xrnidresex 38408 dib0 41166 dicn0 41194 cdlemn11a 41209 dihord6apre 41258 dihatlat 41336 dihpN 41338 eldioph2lem1 42771 eldioph2lem2 42772 dfrtrcl5 43642 dfrcl2 43687 relexpiidm 43717 ushggricedg 47896 uspgrsprfo 48064 rngcidALTV 48190 ringcidALTV 48224 |
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