Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > resiexg | GIF version |
Description: The existence of a restricted identity function, proved without using the Axiom of Replacement. (Contributed by NM, 13-Jan-2007.) |
Ref | Expression |
---|---|
resiexg | ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 4847 | . . 3 ⊢ Rel ( I ↾ 𝐴) | |
2 | simpr 109 | . . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
3 | eleq1 2202 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
4 | 3 | biimpa 294 | . . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) |
5 | 2, 4 | jca 304 | . . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) |
6 | vex 2689 | . . . . . 6 ⊢ 𝑦 ∈ V | |
7 | 6 | opelres 4824 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ ( I ↾ 𝐴) ↔ (〈𝑥, 𝑦〉 ∈ I ∧ 𝑥 ∈ 𝐴)) |
8 | df-br 3930 | . . . . . . 7 ⊢ (𝑥 I 𝑦 ↔ 〈𝑥, 𝑦〉 ∈ I ) | |
9 | 6 | ideq 4691 | . . . . . . 7 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦) |
10 | 8, 9 | bitr3i 185 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 ∈ I ↔ 𝑥 = 𝑦) |
11 | 10 | anbi1i 453 | . . . . 5 ⊢ ((〈𝑥, 𝑦〉 ∈ I ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴)) |
12 | 7, 11 | bitri 183 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ( I ↾ 𝐴) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝐴)) |
13 | opelxp 4569 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) | |
14 | 5, 12, 13 | 3imtr4i 200 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ( I ↾ 𝐴) → 〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐴)) |
15 | 1, 14 | relssi 4630 | . 2 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) |
16 | xpexg 4653 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 × 𝐴) ∈ V) | |
17 | 16 | anidms 394 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × 𝐴) ∈ V) |
18 | ssexg 4067 | . 2 ⊢ ((( I ↾ 𝐴) ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → ( I ↾ 𝐴) ∈ V) | |
19 | 15, 17, 18 | sylancr 410 | 1 ⊢ (𝐴 ∈ 𝑉 → ( I ↾ 𝐴) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1480 Vcvv 2686 ⊆ wss 3071 〈cop 3530 class class class wbr 3929 I cid 4210 × cxp 4537 ↾ cres 4541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-res 4551 |
This theorem is referenced by: ordiso 6921 omct 7002 ctssexmid 7024 ndxarg 11992 subctctexmid 13206 |
Copyright terms: Public domain | W3C validator |