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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 6384). (Contributed by NM, 13-Jan-2007.) |
Ref | Expression |
---|---|
resiexg | ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 5346 | . . 3 ⊢ Rel ( I ↾ 𝐴) | |
2 | simpr 476 | . . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
3 | eleq1 2676 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
4 | 3 | biimpa 500 | . . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) |
5 | 2, 4 | jca 553 | . . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) |
6 | vex 3176 | . . . . . 6 ⊢ 𝑦 ∈ V | |
7 | 6 | opelres 5322 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ ( I ↾ 𝐴) ↔ (〈𝑥, 𝑦〉 ∈ I ∧ 𝑥 ∈ 𝐴)) |
8 | df-br 4584 | . . . . . . 7 ⊢ (𝑥 I 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ I ) | |
9 | 6 | ideq 5196 | . . . . . . 7 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
10 | 8, 9 | bitr3i 265 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ I ↔ 𝑥 = 𝑦) |
11 | 10 | anbi1i 727 | . . . . 5 ⊢ ((〈𝑥, 𝑦〉 ∈ I ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴)) |
12 | 7, 11 | bitri 263 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ( I ↾ 𝐴) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴)) |
13 | opelxp 5070 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) | |
14 | 5, 12, 13 | 3imtr4i 280 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ( I ↾ 𝐴) → 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐴)) |
15 | 1, 14 | relssi 5134 | . 2 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) |
16 | sqxpexg 6861 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V) | |
17 | ssexg 4732 | . 2 ⊢ ((( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → ( I ↾ 𝐴) ∈ V) | |
18 | 15, 16, 17 | sylancr 694 | 1 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 〈cop 4131 class class class wbr 4583 I cid 4948 × cxp 5036 ↾ cres 5040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-res 5050 |
This theorem is referenced by: ordiso 8304 wdomref 8360 dfac9 8841 relexp0g 13610 relexpsucnnr 13613 ndxarg 15715 idfu2nd 16360 idfu1st 16362 idfucl 16364 setcid 16559 equivestrcsetc 16615 pf1ind 19540 islinds2 19971 ausisusgra 25884 cusgraexilem1 25995 sizeusglecusg 26014 poimirlem15 32594 dib0 35471 dicn0 35499 cdlemn11a 35514 dihord6apre 35563 dihatlat 35641 dihpN 35643 eldioph2lem1 36341 eldioph2lem2 36342 dfrtrcl5 36955 dfrcl2 36985 relexpiidm 37015 ausgrusgrb 40395 upgrres1lem1 40528 usgrexi 40661 sizusglecusg 40679 rngcidALTV 41783 ringcidALTV 41846 |
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