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Theorem issref 4650
Description: Two ways to state a relation is reflexive. Adapted from Tarski. (Contributed by FL, 15-Jan-2012.) (Revised by NM, 30-Mar-2016.)
Assertion
Ref Expression
issref (( I ↾ A) ⊆ 𝑅x A x𝑅x)
Distinct variable groups:   x,A   x,𝑅

Proof of Theorem issref
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ral 2305 . 2 (x A x𝑅xx(x Ax𝑅x))
2 vex 2554 . . . . 5 x V
3 opelresi 4566 . . . . 5 (x V → (⟨x, x ( I ↾ A) ↔ x A))
42, 3ax-mp 7 . . . 4 (⟨x, x ( I ↾ A) ↔ x A)
5 df-br 3756 . . . . 5 (x𝑅x ↔ ⟨x, x 𝑅)
65bicomi 123 . . . 4 (⟨x, x 𝑅x𝑅x)
74, 6imbi12i 228 . . 3 ((⟨x, x ( I ↾ A) → ⟨x, x 𝑅) ↔ (x Ax𝑅x))
87albii 1356 . 2 (x(⟨x, x ( I ↾ A) → ⟨x, x 𝑅) ↔ x(x Ax𝑅x))
9 ralidm 3315 . . . . . 6 (x V x V (⟨x, x ( I ↾ A) → ⟨x, x 𝑅) ↔ x V (⟨x, x ( I ↾ A) → ⟨x, x 𝑅))
10 ralv 2565 . . . . . 6 (x V (⟨x, x ( I ↾ A) → ⟨x, x 𝑅) ↔ x(⟨x, x ( I ↾ A) → ⟨x, x 𝑅))
119, 10bitri 173 . . . . 5 (x V x V (⟨x, x ( I ↾ A) → ⟨x, x 𝑅) ↔ x(⟨x, x ( I ↾ A) → ⟨x, x 𝑅))
12 df-ral 2305 . . . . . . . . 9 (x V (⟨x, x ( I ↾ A) → ⟨x, x 𝑅) ↔ x(x V → (⟨x, x ( I ↾ A) → ⟨x, x 𝑅)))
13 pm2.27 35 . . . . . . . . . . . 12 (x V → ((x V → (⟨x, x ( I ↾ A) → ⟨x, x 𝑅)) → (⟨x, x ( I ↾ A) → ⟨x, x 𝑅)))
14 opelresg 4562 . . . . . . . . . . . . . . 15 (z V → (⟨x, z ( I ↾ A) ↔ (⟨x, z I x A)))
15 df-br 3756 . . . . . . . . . . . . . . . . 17 (x I z ↔ ⟨x, z I )
16 vex 2554 . . . . . . . . . . . . . . . . . . 19 z V
1716ideq 4431 . . . . . . . . . . . . . . . . . 18 (x I zx = z)
18 opelresi 4566 . . . . . . . . . . . . . . . . . . . . 21 (x A → (⟨x, x ( I ↾ A) ↔ x A))
19 pm2.27 35 . . . . . . . . . . . . . . . . . . . . . 22 (⟨x, x ( I ↾ A) → ((⟨x, x ( I ↾ A) → ⟨x, x 𝑅) → ⟨x, x 𝑅))
20 opeq2 3541 . . . . . . . . . . . . . . . . . . . . . . . 24 (x = z → ⟨x, x⟩ = ⟨x, z⟩)
2120eleq1d 2103 . . . . . . . . . . . . . . . . . . . . . . 23 (x = z → (⟨x, x 𝑅 ↔ ⟨x, z 𝑅))
2221biimpcd 148 . . . . . . . . . . . . . . . . . . . . . 22 (⟨x, x 𝑅 → (x = z → ⟨x, z 𝑅))
2319, 22syl6 29 . . . . . . . . . . . . . . . . . . . . 21 (⟨x, x ( I ↾ A) → ((⟨x, x ( I ↾ A) → ⟨x, x 𝑅) → (x = z → ⟨x, z 𝑅)))
2418, 23syl6bir 153 . . . . . . . . . . . . . . . . . . . 20 (x A → (x A → ((⟨x, x ( I ↾ A) → ⟨x, x 𝑅) → (x = z → ⟨x, z 𝑅))))
2524pm2.43i 43 . . . . . . . . . . . . . . . . . . 19 (x A → ((⟨x, x ( I ↾ A) → ⟨x, x 𝑅) → (x = z → ⟨x, z 𝑅)))
2625com3r 73 . . . . . . . . . . . . . . . . . 18 (x = z → (x A → ((⟨x, x ( I ↾ A) → ⟨x, x 𝑅) → ⟨x, z 𝑅)))
2717, 26sylbi 114 . . . . . . . . . . . . . . . . 17 (x I z → (x A → ((⟨x, x ( I ↾ A) → ⟨x, x 𝑅) → ⟨x, z 𝑅)))
2815, 27sylbir 125 . . . . . . . . . . . . . . . 16 (⟨x, z I → (x A → ((⟨x, x ( I ↾ A) → ⟨x, x 𝑅) → ⟨x, z 𝑅)))
2928imp 115 . . . . . . . . . . . . . . 15 ((⟨x, z I x A) → ((⟨x, x ( I ↾ A) → ⟨x, x 𝑅) → ⟨x, z 𝑅))
3014, 29syl6bi 152 . . . . . . . . . . . . . 14 (z V → (⟨x, z ( I ↾ A) → ((⟨x, x ( I ↾ A) → ⟨x, x 𝑅) → ⟨x, z 𝑅)))
3130com3r 73 . . . . . . . . . . . . 13 ((⟨x, x ( I ↾ A) → ⟨x, x 𝑅) → (z V → (⟨x, z ( I ↾ A) → ⟨x, z 𝑅)))
3231ralrimiv 2385 . . . . . . . . . . . 12 ((⟨x, x ( I ↾ A) → ⟨x, x 𝑅) → z V (⟨x, z ( I ↾ A) → ⟨x, z 𝑅))
3313, 32syl6 29 . . . . . . . . . . 11 (x V → ((x V → (⟨x, x ( I ↾ A) → ⟨x, x 𝑅)) → z V (⟨x, z ( I ↾ A) → ⟨x, z 𝑅)))
342, 33ax-mp 7 . . . . . . . . . 10 ((x V → (⟨x, x ( I ↾ A) → ⟨x, x 𝑅)) → z V (⟨x, z ( I ↾ A) → ⟨x, z 𝑅))
3534sps 1427 . . . . . . . . 9 (x(x V → (⟨x, x ( I ↾ A) → ⟨x, x 𝑅)) → z V (⟨x, z ( I ↾ A) → ⟨x, z 𝑅))
3612, 35sylbi 114 . . . . . . . 8 (x V (⟨x, x ( I ↾ A) → ⟨x, x 𝑅) → z V (⟨x, z ( I ↾ A) → ⟨x, z 𝑅))
3736ralimi 2378 . . . . . . 7 (x V x V (⟨x, x ( I ↾ A) → ⟨x, x 𝑅) → x V z V (⟨x, z ( I ↾ A) → ⟨x, z 𝑅))
38 eleq1 2097 . . . . . . . . 9 (y = ⟨x, z⟩ → (y ( I ↾ A) ↔ ⟨x, z ( I ↾ A)))
39 eleq1 2097 . . . . . . . . 9 (y = ⟨x, z⟩ → (y 𝑅 ↔ ⟨x, z 𝑅))
4038, 39imbi12d 223 . . . . . . . 8 (y = ⟨x, z⟩ → ((y ( I ↾ A) → y 𝑅) ↔ (⟨x, z ( I ↾ A) → ⟨x, z 𝑅)))
4140ralxp 4422 . . . . . . 7 (y (V × V)(y ( I ↾ A) → y 𝑅) ↔ x V z V (⟨x, z ( I ↾ A) → ⟨x, z 𝑅))
4237, 41sylibr 137 . . . . . 6 (x V x V (⟨x, x ( I ↾ A) → ⟨x, x 𝑅) → y (V × V)(y ( I ↾ A) → y 𝑅))
43 df-ral 2305 . . . . . . 7 (y (V × V)(y ( I ↾ A) → y 𝑅) ↔ y(y (V × V) → (y ( I ↾ A) → y 𝑅)))
44 relres 4582 . . . . . . . . . . . 12 Rel ( I ↾ A)
45 df-rel 4295 . . . . . . . . . . . 12 (Rel ( I ↾ A) ↔ ( I ↾ A) ⊆ (V × V))
4644, 45mpbi 133 . . . . . . . . . . 11 ( I ↾ A) ⊆ (V × V)
4746sseli 2935 . . . . . . . . . 10 (y ( I ↾ A) → y (V × V))
4847ancri 307 . . . . . . . . 9 (y ( I ↾ A) → (y (V × V) y ( I ↾ A)))
49 pm3.31 249 . . . . . . . . 9 ((y (V × V) → (y ( I ↾ A) → y 𝑅)) → ((y (V × V) y ( I ↾ A)) → y 𝑅))
5048, 49syl5 28 . . . . . . . 8 ((y (V × V) → (y ( I ↾ A) → y 𝑅)) → (y ( I ↾ A) → y 𝑅))
5150alimi 1341 . . . . . . 7 (y(y (V × V) → (y ( I ↾ A) → y 𝑅)) → y(y ( I ↾ A) → y 𝑅))
5243, 51sylbi 114 . . . . . 6 (y (V × V)(y ( I ↾ A) → y 𝑅) → y(y ( I ↾ A) → y 𝑅))
5342, 52syl 14 . . . . 5 (x V x V (⟨x, x ( I ↾ A) → ⟨x, x 𝑅) → y(y ( I ↾ A) → y 𝑅))
5411, 53sylbir 125 . . . 4 (x(⟨x, x ( I ↾ A) → ⟨x, x 𝑅) → y(y ( I ↾ A) → y 𝑅))
55 dfss2 2928 . . . 4 (( I ↾ A) ⊆ 𝑅y(y ( I ↾ A) → y 𝑅))
5654, 55sylibr 137 . . 3 (x(⟨x, x ( I ↾ A) → ⟨x, x 𝑅) → ( I ↾ A) ⊆ 𝑅)
57 ssel 2933 . . . 4 (( I ↾ A) ⊆ 𝑅 → (⟨x, x ( I ↾ A) → ⟨x, x 𝑅))
5857alrimiv 1751 . . 3 (( I ↾ A) ⊆ 𝑅x(⟨x, x ( I ↾ A) → ⟨x, x 𝑅))
5956, 58impbii 117 . 2 (x(⟨x, x ( I ↾ A) → ⟨x, x 𝑅) ↔ ( I ↾ A) ⊆ 𝑅)
601, 8, 593bitr2ri 198 1 (( I ↾ A) ⊆ 𝑅x A x𝑅x)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242   wcel 1390  wral 2300  Vcvv 2551  wss 2911  cop 3370   class class class wbr 3755   I cid 4016   × cxp 4286  cres 4290  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-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-iun 3650  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-res 4300
This theorem is referenced by: (None)
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