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Theorem brtpos2 5807
Description: Value of the transposition at a pair A, B. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
brtpos2 (B 𝑉 → (Atpos 𝐹B ↔ (A (dom 𝐹 ∪ {∅}) {A}𝐹B)))

Proof of Theorem brtpos2
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reltpos 5806 . . . 4 Rel tpos 𝐹
21brrelexi 4327 . . 3 (Atpos 𝐹BA V)
32a1i 9 . 2 (B 𝑉 → (Atpos 𝐹BA V))
4 elex 2560 . . . 4 (A (dom 𝐹 ∪ {∅}) → A V)
54adantr 261 . . 3 ((A (dom 𝐹 ∪ {∅}) {A}𝐹B) → A V)
65a1i 9 . 2 (B 𝑉 → ((A (dom 𝐹 ∪ {∅}) {A}𝐹B) → A V))
7 df-tpos 5801 . . . . . 6 tpos 𝐹 = (𝐹 ∘ (x (dom 𝐹 ∪ {∅}) ↦ {x}))
87breqi 3761 . . . . 5 (Atpos 𝐹BA(𝐹 ∘ (x (dom 𝐹 ∪ {∅}) ↦ {x}))B)
9 brcog 4445 . . . . 5 ((A V B 𝑉) → (A(𝐹 ∘ (x (dom 𝐹 ∪ {∅}) ↦ {x}))By(A(x (dom 𝐹 ∪ {∅}) ↦ {x})y y𝐹B)))
108, 9syl5bb 181 . . . 4 ((A V B 𝑉) → (Atpos 𝐹By(A(x (dom 𝐹 ∪ {∅}) ↦ {x})y y𝐹B)))
11 funmpt 4881 . . . . . . . . . . 11 Fun (x (dom 𝐹 ∪ {∅}) ↦ {x})
12 funbrfv2b 5161 . . . . . . . . . . 11 (Fun (x (dom 𝐹 ∪ {∅}) ↦ {x}) → (A(x (dom 𝐹 ∪ {∅}) ↦ {x})y ↔ (A dom (x (dom 𝐹 ∪ {∅}) ↦ {x}) ((x (dom 𝐹 ∪ {∅}) ↦ {x})‘A) = y)))
1311, 12ax-mp 7 . . . . . . . . . 10 (A(x (dom 𝐹 ∪ {∅}) ↦ {x})y ↔ (A dom (x (dom 𝐹 ∪ {∅}) ↦ {x}) ((x (dom 𝐹 ∪ {∅}) ↦ {x})‘A) = y))
14 vex 2554 . . . . . . . . . . . . . . . . 17 x V
15 snexg 3927 . . . . . . . . . . . . . . . . 17 (x V → {x} V)
1614, 15ax-mp 7 . . . . . . . . . . . . . . . 16 {x} V
1716cnvex 4799 . . . . . . . . . . . . . . 15 {x} V
1817uniex 4140 . . . . . . . . . . . . . 14 {x} V
19 eqid 2037 . . . . . . . . . . . . . 14 (x (dom 𝐹 ∪ {∅}) ↦ {x}) = (x (dom 𝐹 ∪ {∅}) ↦ {x})
2018, 19dmmpti 4971 . . . . . . . . . . . . 13 dom (x (dom 𝐹 ∪ {∅}) ↦ {x}) = (dom 𝐹 ∪ {∅})
2120eleq2i 2101 . . . . . . . . . . . 12 (A dom (x (dom 𝐹 ∪ {∅}) ↦ {x}) ↔ A (dom 𝐹 ∪ {∅}))
22 eqcom 2039 . . . . . . . . . . . 12 (((x (dom 𝐹 ∪ {∅}) ↦ {x})‘A) = yy = ((x (dom 𝐹 ∪ {∅}) ↦ {x})‘A))
2321, 22anbi12i 433 . . . . . . . . . . 11 ((A dom (x (dom 𝐹 ∪ {∅}) ↦ {x}) ((x (dom 𝐹 ∪ {∅}) ↦ {x})‘A) = y) ↔ (A (dom 𝐹 ∪ {∅}) y = ((x (dom 𝐹 ∪ {∅}) ↦ {x})‘A)))
24 snexg 3927 . . . . . . . . . . . . . . . 16 (A (dom 𝐹 ∪ {∅}) → {A} V)
25 cnvexg 4798 . . . . . . . . . . . . . . . 16 ({A} V → {A} V)
2624, 25syl 14 . . . . . . . . . . . . . . 15 (A (dom 𝐹 ∪ {∅}) → {A} V)
27 uniexg 4141 . . . . . . . . . . . . . . 15 ({A} V → {A} V)
2826, 27syl 14 . . . . . . . . . . . . . 14 (A (dom 𝐹 ∪ {∅}) → {A} V)
29 sneq 3378 . . . . . . . . . . . . . . . . 17 (x = A → {x} = {A})
3029cnveqd 4454 . . . . . . . . . . . . . . . 16 (x = A{x} = {A})
3130unieqd 3582 . . . . . . . . . . . . . . 15 (x = A {x} = {A})
3231, 19fvmptg 5191 . . . . . . . . . . . . . 14 ((A (dom 𝐹 ∪ {∅}) {A} V) → ((x (dom 𝐹 ∪ {∅}) ↦ {x})‘A) = {A})
3328, 32mpdan 398 . . . . . . . . . . . . 13 (A (dom 𝐹 ∪ {∅}) → ((x (dom 𝐹 ∪ {∅}) ↦ {x})‘A) = {A})
3433eqeq2d 2048 . . . . . . . . . . . 12 (A (dom 𝐹 ∪ {∅}) → (y = ((x (dom 𝐹 ∪ {∅}) ↦ {x})‘A) ↔ y = {A}))
3534pm5.32i 427 . . . . . . . . . . 11 ((A (dom 𝐹 ∪ {∅}) y = ((x (dom 𝐹 ∪ {∅}) ↦ {x})‘A)) ↔ (A (dom 𝐹 ∪ {∅}) y = {A}))
3623, 35bitri 173 . . . . . . . . . 10 ((A dom (x (dom 𝐹 ∪ {∅}) ↦ {x}) ((x (dom 𝐹 ∪ {∅}) ↦ {x})‘A) = y) ↔ (A (dom 𝐹 ∪ {∅}) y = {A}))
3713, 36bitri 173 . . . . . . . . 9 (A(x (dom 𝐹 ∪ {∅}) ↦ {x})y ↔ (A (dom 𝐹 ∪ {∅}) y = {A}))
38 ancom 253 . . . . . . . . 9 ((A (dom 𝐹 ∪ {∅}) y = {A}) ↔ (y = {A} A (dom 𝐹 ∪ {∅})))
3937, 38bitri 173 . . . . . . . 8 (A(x (dom 𝐹 ∪ {∅}) ↦ {x})y ↔ (y = {A} A (dom 𝐹 ∪ {∅})))
4039anbi1i 431 . . . . . . 7 ((A(x (dom 𝐹 ∪ {∅}) ↦ {x})y y𝐹B) ↔ ((y = {A} A (dom 𝐹 ∪ {∅})) y𝐹B))
41 anass 381 . . . . . . 7 (((y = {A} A (dom 𝐹 ∪ {∅})) y𝐹B) ↔ (y = {A} (A (dom 𝐹 ∪ {∅}) y𝐹B)))
4240, 41bitri 173 . . . . . 6 ((A(x (dom 𝐹 ∪ {∅}) ↦ {x})y y𝐹B) ↔ (y = {A} (A (dom 𝐹 ∪ {∅}) y𝐹B)))
4342exbii 1493 . . . . 5 (y(A(x (dom 𝐹 ∪ {∅}) ↦ {x})y y𝐹B) ↔ y(y = {A} (A (dom 𝐹 ∪ {∅}) y𝐹B)))
44 exsimpr 1506 . . . . . . 7 (y(y = {A} (A (dom 𝐹 ∪ {∅}) y𝐹B)) → y(A (dom 𝐹 ∪ {∅}) y𝐹B))
45 exsimpl 1505 . . . . . . . 8 (y(A (dom 𝐹 ∪ {∅}) y𝐹B) → y A (dom 𝐹 ∪ {∅}))
46 19.9v 1748 . . . . . . . 8 (y A (dom 𝐹 ∪ {∅}) ↔ A (dom 𝐹 ∪ {∅}))
4745, 46sylib 127 . . . . . . 7 (y(A (dom 𝐹 ∪ {∅}) y𝐹B) → A (dom 𝐹 ∪ {∅}))
4844, 47syl 14 . . . . . 6 (y(y = {A} (A (dom 𝐹 ∪ {∅}) y𝐹B)) → A (dom 𝐹 ∪ {∅}))
49 simpl 102 . . . . . 6 ((A (dom 𝐹 ∪ {∅}) {A}𝐹B) → A (dom 𝐹 ∪ {∅}))
50 breq1 3758 . . . . . . . . 9 (y = {A} → (y𝐹B {A}𝐹B))
5150anbi2d 437 . . . . . . . 8 (y = {A} → ((A (dom 𝐹 ∪ {∅}) y𝐹B) ↔ (A (dom 𝐹 ∪ {∅}) {A}𝐹B)))
5251ceqsexgv 2667 . . . . . . 7 ( {A} V → (y(y = {A} (A (dom 𝐹 ∪ {∅}) y𝐹B)) ↔ (A (dom 𝐹 ∪ {∅}) {A}𝐹B)))
5328, 52syl 14 . . . . . 6 (A (dom 𝐹 ∪ {∅}) → (y(y = {A} (A (dom 𝐹 ∪ {∅}) y𝐹B)) ↔ (A (dom 𝐹 ∪ {∅}) {A}𝐹B)))
5448, 49, 53pm5.21nii 619 . . . . 5 (y(y = {A} (A (dom 𝐹 ∪ {∅}) y𝐹B)) ↔ (A (dom 𝐹 ∪ {∅}) {A}𝐹B))
5543, 54bitri 173 . . . 4 (y(A(x (dom 𝐹 ∪ {∅}) ↦ {x})y y𝐹B) ↔ (A (dom 𝐹 ∪ {∅}) {A}𝐹B))
5610, 55syl6bb 185 . . 3 ((A V B 𝑉) → (Atpos 𝐹B ↔ (A (dom 𝐹 ∪ {∅}) {A}𝐹B)))
5756expcom 109 . 2 (B 𝑉 → (A V → (Atpos 𝐹B ↔ (A (dom 𝐹 ∪ {∅}) {A}𝐹B))))
583, 6, 57pm5.21ndd 620 1 (B 𝑉 → (Atpos 𝐹B ↔ (A (dom 𝐹 ∪ {∅}) {A}𝐹B)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  cun 2909  c0 3218  {csn 3367   cuni 3571   class class class wbr 3755  cmpt 3809  ccnv 4287  dom cdm 4288  ccom 4292  Fun wfun 4839  cfv 4845  tpos ctpos 5800
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-13 1401  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  ax-un 4136
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-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853  df-tpos 5801
This theorem is referenced by:  brtpos0  5808  reldmtpos  5809  brtposg  5810  dftpos4  5819  tpostpos  5820
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