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Mirrors > Home > ILE Home > Th. List > coi1 | GIF version |
Description: Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.) |
Ref | Expression |
---|---|
coi1 | ⊢ (Rel A → (A ∘ I ) = A) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relco 4762 | . 2 ⊢ Rel (A ∘ I ) | |
2 | vex 2554 | . . . . . 6 ⊢ x ∈ V | |
3 | vex 2554 | . . . . . 6 ⊢ y ∈ V | |
4 | 2, 3 | opelco 4450 | . . . . 5 ⊢ (〈x, y〉 ∈ (A ∘ I ) ↔ ∃z(x I z ∧ zAy)) |
5 | vex 2554 | . . . . . . . . . 10 ⊢ z ∈ V | |
6 | 5 | ideq 4431 | . . . . . . . . 9 ⊢ (x I z ↔ x = z) |
7 | equcom 1590 | . . . . . . . . 9 ⊢ (x = z ↔ z = x) | |
8 | 6, 7 | bitri 173 | . . . . . . . 8 ⊢ (x I z ↔ z = x) |
9 | 8 | anbi1i 431 | . . . . . . 7 ⊢ ((x I z ∧ zAy) ↔ (z = x ∧ zAy)) |
10 | 9 | exbii 1493 | . . . . . 6 ⊢ (∃z(x I z ∧ zAy) ↔ ∃z(z = x ∧ zAy)) |
11 | breq1 3758 | . . . . . . 7 ⊢ (z = x → (zAy ↔ xAy)) | |
12 | 2, 11 | ceqsexv 2587 | . . . . . 6 ⊢ (∃z(z = x ∧ zAy) ↔ xAy) |
13 | 10, 12 | bitri 173 | . . . . 5 ⊢ (∃z(x I z ∧ zAy) ↔ xAy) |
14 | 4, 13 | bitri 173 | . . . 4 ⊢ (〈x, y〉 ∈ (A ∘ I ) ↔ xAy) |
15 | df-br 3756 | . . . 4 ⊢ (xAy ↔ 〈x, y〉 ∈ A) | |
16 | 14, 15 | bitri 173 | . . 3 ⊢ (〈x, y〉 ∈ (A ∘ I ) ↔ 〈x, y〉 ∈ A) |
17 | 16 | eqrelriv 4376 | . 2 ⊢ ((Rel (A ∘ I ) ∧ Rel A) → (A ∘ I ) = A) |
18 | 1, 17 | mpan 400 | 1 ⊢ (Rel A → (A ∘ I ) = A) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ∃wex 1378 ∈ wcel 1390 〈cop 3370 class class class wbr 3755 I cid 4016 ∘ ccom 4292 Rel wrel 4293 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-co 4297 |
This theorem is referenced by: coi2 4780 coires1 4781 relcoi1 4792 fcoi1 5013 |
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