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Theorem brtpos2 5896
Description: Value of the transposition at a pair 𝐴, 𝐵. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
brtpos2 (𝐵𝑉 → (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 5895 . . . 4 Rel tpos 𝐹
21brrelexi 4411 . . 3 (𝐴tpos 𝐹𝐵𝐴 ∈ V)
32a1i 9 . 2 (𝐵𝑉 → (𝐴tpos 𝐹𝐵𝐴 ∈ V))
4 elex 2583 . . . 4 (𝐴 ∈ (dom 𝐹 ∪ {∅}) → 𝐴 ∈ V)
54adantr 265 . . 3 ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵) → 𝐴 ∈ V)
65a1i 9 . 2 (𝐵𝑉 → ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵) → 𝐴 ∈ V))
7 df-tpos 5890 . . . . . 6 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
87breqi 3797 . . . . 5 (𝐴tpos 𝐹𝐵𝐴(𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))𝐵)
9 brcog 4529 . . . . 5 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴(𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))𝐵 ↔ ∃𝑦(𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵)))
108, 9syl5bb 185 . . . 4 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴tpos 𝐹𝐵 ↔ ∃𝑦(𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵)))
11 funmpt 4965 . . . . . . . . . . 11 Fun (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})
12 funbrfv2b 5245 . . . . . . . . . . 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 2577 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
15 snexg 3963 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ V → {𝑥} ∈ V)
1614, 15ax-mp 7 . . . . . . . . . . . . . . . 16 {𝑥} ∈ V
1716cnvex 4883 . . . . . . . . . . . . . . 15 {𝑥} ∈ V
1817uniex 4201 . . . . . . . . . . . . . 14 {𝑥} ∈ V
19 eqid 2056 . . . . . . . . . . . . . 14 (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) = (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})
2018, 19dmmpti 5055 . . . . . . . . . . . . 13 dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) = (dom 𝐹 ∪ {∅})
2120eleq2i 2120 . . . . . . . . . . . 12 (𝐴 ∈ dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ↔ 𝐴 ∈ (dom 𝐹 ∪ {∅}))
22 eqcom 2058 . . . . . . . . . . . 12 (((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) = 𝑦𝑦 = ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴))
2321, 22anbi12i 441 . . . . . . . . . . 11 ((𝐴 ∈ dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ∧ ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) = 𝑦) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴)))
24 snexg 3963 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (dom 𝐹 ∪ {∅}) → {𝐴} ∈ V)
25 cnvexg 4882 . . . . . . . . . . . . . . . 16 ({𝐴} ∈ V → {𝐴} ∈ V)
2624, 25syl 14 . . . . . . . . . . . . . . 15 (𝐴 ∈ (dom 𝐹 ∪ {∅}) → {𝐴} ∈ V)
27 uniexg 4202 . . . . . . . . . . . . . . 15 ({𝐴} ∈ V → {𝐴} ∈ V)
2826, 27syl 14 . . . . . . . . . . . . . 14 (𝐴 ∈ (dom 𝐹 ∪ {∅}) → {𝐴} ∈ V)
29 sneq 3413 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐴 → {𝑥} = {𝐴})
3029cnveqd 4538 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐴{𝑥} = {𝐴})
3130unieqd 3618 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴 {𝑥} = {𝐴})
3231, 19fvmptg 5275 . . . . . . . . . . . . . 14 ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴} ∈ V) → ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) = {𝐴})
3328, 32mpdan 406 . . . . . . . . . . . . 13 (𝐴 ∈ (dom 𝐹 ∪ {∅}) → ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) = {𝐴})
3433eqeq2d 2067 . . . . . . . . . . . 12 (𝐴 ∈ (dom 𝐹 ∪ {∅}) → (𝑦 = ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) ↔ 𝑦 = {𝐴}))
3534pm5.32i 435 . . . . . . . . . . 11 ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴)) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = {𝐴}))
3623, 35bitri 177 . . . . . . . . . 10 ((𝐴 ∈ dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ∧ ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) = 𝑦) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = {𝐴}))
3713, 36bitri 177 . . . . . . . . 9 (𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦 ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = {𝐴}))
38 ancom 257 . . . . . . . . 9 ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = {𝐴}) ↔ (𝑦 = {𝐴} ∧ 𝐴 ∈ (dom 𝐹 ∪ {∅})))
3937, 38bitri 177 . . . . . . . 8 (𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦 ↔ (𝑦 = {𝐴} ∧ 𝐴 ∈ (dom 𝐹 ∪ {∅})))
4039anbi1i 439 . . . . . . 7 ((𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵) ↔ ((𝑦 = {𝐴} ∧ 𝐴 ∈ (dom 𝐹 ∪ {∅})) ∧ 𝑦𝐹𝐵))
41 anass 387 . . . . . . 7 (((𝑦 = {𝐴} ∧ 𝐴 ∈ (dom 𝐹 ∪ {∅})) ∧ 𝑦𝐹𝐵) ↔ (𝑦 = {𝐴} ∧ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
4240, 41bitri 177 . . . . . 6 ((𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵) ↔ (𝑦 = {𝐴} ∧ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
4342exbii 1512 . . . . 5 (∃𝑦(𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵) ↔ ∃𝑦(𝑦 = {𝐴} ∧ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
44 exsimpr 1525 . . . . . . 7 (∃𝑦(𝑦 = {𝐴} ∧ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)) → ∃𝑦(𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵))
45 exsimpl 1524 . . . . . . . 8 (∃𝑦(𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵) → ∃𝑦 𝐴 ∈ (dom 𝐹 ∪ {∅}))
46 19.9v 1767 . . . . . . . 8 (∃𝑦 𝐴 ∈ (dom 𝐹 ∪ {∅}) ↔ 𝐴 ∈ (dom 𝐹 ∪ {∅}))
4745, 46sylib 131 . . . . . . 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 106 . . . . . 6 ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵) → 𝐴 ∈ (dom 𝐹 ∪ {∅}))
50 breq1 3794 . . . . . . . . 9 (𝑦 = {𝐴} → (𝑦𝐹𝐵 {𝐴}𝐹𝐵))
5150anbi2d 445 . . . . . . . 8 (𝑦 = {𝐴} → ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵)))
5251ceqsexgv 2695 . . . . . . 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 630 . . . . 5 (∃𝑦(𝑦 = {𝐴} ∧ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵))
5543, 54bitri 177 . . . 4 (∃𝑦(𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵))
5610, 55syl6bb 189 . . 3 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵)))
5756expcom 113 . 2 (𝐵𝑉 → (𝐴 ∈ V → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵))))
583, 6, 57pm5.21ndd 631 1 (𝐵𝑉 → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wex 1397  wcel 1409  Vcvv 2574  cun 2942  c0 3251  {csn 3402   cuni 3607   class class class wbr 3791  cmpt 3845  ccnv 4371  dom cdm 4372  ccom 4376  Fun wfun 4923  cfv 4929  tpos ctpos 5889
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-fv 4937  df-tpos 5890
This theorem is referenced by:  brtpos0  5897  reldmtpos  5898  brtposg  5899  dftpos4  5908  tpostpos  5909
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