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Theorem brtpos2 7887
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 7886 . . . 4 Rel tpos 𝐹
21brrelex1i 5601 . . 3 (𝐴tpos 𝐹𝐵𝐴 ∈ V)
32a1i 11 . 2 (𝐵𝑉 → (𝐴tpos 𝐹𝐵𝐴 ∈ V))
4 elex 3510 . . . 4 (𝐴 ∈ (dom 𝐹 ∪ {∅}) → 𝐴 ∈ V)
54adantr 481 . . 3 ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵) → 𝐴 ∈ V)
65a1i 11 . 2 (𝐵𝑉 → ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵) → 𝐴 ∈ V))
7 df-tpos 7881 . . . . . 6 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
87breqi 5063 . . . . 5 (𝐴tpos 𝐹𝐵𝐴(𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))𝐵)
9 brcog 5730 . . . . 5 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴(𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))𝐵 ↔ ∃𝑦(𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵)))
108, 9syl5bb 284 . . . 4 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴tpos 𝐹𝐵 ↔ ∃𝑦(𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵)))
11 funmpt 6386 . . . . . . . . . . 11 Fun (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})
12 funbrfv2b 6716 . . . . . . . . . . 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 5 . . . . . . . . . 10 (𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦 ↔ (𝐴 ∈ dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ∧ ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) = 𝑦))
14 snex 5322 . . . . . . . . . . . . . . . 16 {𝑥} ∈ V
1514cnvex 7619 . . . . . . . . . . . . . . 15 {𝑥} ∈ V
1615uniex 7454 . . . . . . . . . . . . . 14 {𝑥} ∈ V
17 eqid 2818 . . . . . . . . . . . . . 14 (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) = (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})
1816, 17dmmpti 6485 . . . . . . . . . . . . 13 dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) = (dom 𝐹 ∪ {∅})
1918eleq2i 2901 . . . . . . . . . . . 12 (𝐴 ∈ dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ↔ 𝐴 ∈ (dom 𝐹 ∪ {∅}))
20 eqcom 2825 . . . . . . . . . . . 12 (((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) = 𝑦𝑦 = ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴))
2119, 20anbi12i 626 . . . . . . . . . . 11 ((𝐴 ∈ dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ∧ ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) = 𝑦) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴)))
22 sneq 4567 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐴 → {𝑥} = {𝐴})
2322cnveqd 5739 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴{𝑥} = {𝐴})
2423unieqd 4840 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 {𝑥} = {𝐴})
25 snex 5322 . . . . . . . . . . . . . . . 16 {𝐴} ∈ V
2625cnvex 7619 . . . . . . . . . . . . . . 15 {𝐴} ∈ V
2726uniex 7454 . . . . . . . . . . . . . 14 {𝐴} ∈ V
2824, 17, 27fvmpt 6761 . . . . . . . . . . . . 13 (𝐴 ∈ (dom 𝐹 ∪ {∅}) → ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) = {𝐴})
2928eqeq2d 2829 . . . . . . . . . . . 12 (𝐴 ∈ (dom 𝐹 ∪ {∅}) → (𝑦 = ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) ↔ 𝑦 = {𝐴}))
3029pm5.32i 575 . . . . . . . . . . 11 ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴)) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = {𝐴}))
3121, 30bitri 276 . . . . . . . . . 10 ((𝐴 ∈ dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ∧ ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) = 𝑦) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = {𝐴}))
3213, 31bitri 276 . . . . . . . . 9 (𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦 ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = {𝐴}))
3332biancomi 463 . . . . . . . 8 (𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦 ↔ (𝑦 = {𝐴} ∧ 𝐴 ∈ (dom 𝐹 ∪ {∅})))
3433anbi1i 623 . . . . . . 7 ((𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵) ↔ ((𝑦 = {𝐴} ∧ 𝐴 ∈ (dom 𝐹 ∪ {∅})) ∧ 𝑦𝐹𝐵))
35 anass 469 . . . . . . 7 (((𝑦 = {𝐴} ∧ 𝐴 ∈ (dom 𝐹 ∪ {∅})) ∧ 𝑦𝐹𝐵) ↔ (𝑦 = {𝐴} ∧ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
3634, 35bitri 276 . . . . . 6 ((𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵) ↔ (𝑦 = {𝐴} ∧ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
3736exbii 1839 . . . . 5 (∃𝑦(𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵) ↔ ∃𝑦(𝑦 = {𝐴} ∧ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
38 breq1 5060 . . . . . . 7 (𝑦 = {𝐴} → (𝑦𝐹𝐵 {𝐴}𝐹𝐵))
3938anbi2d 628 . . . . . 6 (𝑦 = {𝐴} → ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵)))
4027, 39ceqsexv 3539 . . . . 5 (∃𝑦(𝑦 = {𝐴} ∧ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵))
4137, 40bitri 276 . . . 4 (∃𝑦(𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵))
4210, 41syl6bb 288 . . 3 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵)))
4342expcom 414 . 2 (𝐵𝑉 → (𝐴 ∈ V → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵))))
443, 6, 43pm5.21ndd 381 1 (𝐵𝑉 → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wex 1771  wcel 2105  Vcvv 3492  cun 3931  c0 4288  {csn 4557   cuni 4830   class class class wbr 5057  cmpt 5137  ccnv 5547  dom cdm 5548  ccom 5552  Fun wfun 6342  cfv 6348  tpos ctpos 7880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356  df-tpos 7881
This theorem is referenced by:  brtpos0  7888  reldmtpos  7889  brtpos  7890  dftpos4  7900  tpostpos  7901
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