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Theorem brtpos2 7885
 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 7884 . . . 4 Rel tpos 𝐹
21brrelex1i 5576 . . 3 (𝐴tpos 𝐹𝐵𝐴 ∈ V)
32a1i 11 . 2 (𝐵𝑉 → (𝐴tpos 𝐹𝐵𝐴 ∈ V))
4 elex 3462 . . . 4 (𝐴 ∈ (dom 𝐹 ∪ {∅}) → 𝐴 ∈ V)
54adantr 484 . . 3 ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵) → 𝐴 ∈ V)
65a1i 11 . 2 (𝐵𝑉 → ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵) → 𝐴 ∈ V))
7 df-tpos 7879 . . . . . 6 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
87breqi 5039 . . . . 5 (𝐴tpos 𝐹𝐵𝐴(𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))𝐵)
9 brcog 5705 . . . . 5 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴(𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))𝐵 ↔ ∃𝑦(𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵)))
108, 9syl5bb 286 . . . 4 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴tpos 𝐹𝐵 ↔ ∃𝑦(𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵)))
11 funmpt 6366 . . . . . . . . . . 11 Fun (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})
12 funbrfv2b 6702 . . . . . . . . . . 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 5300 . . . . . . . . . . . . . . . 16 {𝑥} ∈ V
1514cnvex 7616 . . . . . . . . . . . . . . 15 {𝑥} ∈ V
1615uniex 7451 . . . . . . . . . . . . . 14 {𝑥} ∈ V
17 eqid 2801 . . . . . . . . . . . . . 14 (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) = (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})
1816, 17dmmpti 6468 . . . . . . . . . . . . 13 dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) = (dom 𝐹 ∪ {∅})
1918eleq2i 2884 . . . . . . . . . . . 12 (𝐴 ∈ dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ↔ 𝐴 ∈ (dom 𝐹 ∪ {∅}))
20 eqcom 2808 . . . . . . . . . . . 12 (((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) = 𝑦𝑦 = ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴))
2119, 20anbi12i 629 . . . . . . . . . . 11 ((𝐴 ∈ dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ∧ ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) = 𝑦) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴)))
22 sneq 4538 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐴 → {𝑥} = {𝐴})
2322cnveqd 5714 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴{𝑥} = {𝐴})
2423unieqd 4817 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 {𝑥} = {𝐴})
25 snex 5300 . . . . . . . . . . . . . . . 16 {𝐴} ∈ V
2625cnvex 7616 . . . . . . . . . . . . . . 15 {𝐴} ∈ V
2726uniex 7451 . . . . . . . . . . . . . 14 {𝐴} ∈ V
2824, 17, 27fvmpt 6749 . . . . . . . . . . . . 13 (𝐴 ∈ (dom 𝐹 ∪ {∅}) → ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) = {𝐴})
2928eqeq2d 2812 . . . . . . . . . . . 12 (𝐴 ∈ (dom 𝐹 ∪ {∅}) → (𝑦 = ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) ↔ 𝑦 = {𝐴}))
3029pm5.32i 578 . . . . . . . . . . 11 ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴)) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = {𝐴}))
3121, 30bitri 278 . . . . . . . . . 10 ((𝐴 ∈ dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ∧ ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) = 𝑦) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = {𝐴}))
3213, 31bitri 278 . . . . . . . . 9 (𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦 ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = {𝐴}))
3332biancomi 466 . . . . . . . 8 (𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦 ↔ (𝑦 = {𝐴} ∧ 𝐴 ∈ (dom 𝐹 ∪ {∅})))
3433anbi1i 626 . . . . . . 7 ((𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵) ↔ ((𝑦 = {𝐴} ∧ 𝐴 ∈ (dom 𝐹 ∪ {∅})) ∧ 𝑦𝐹𝐵))
35 anass 472 . . . . . . 7 (((𝑦 = {𝐴} ∧ 𝐴 ∈ (dom 𝐹 ∪ {∅})) ∧ 𝑦𝐹𝐵) ↔ (𝑦 = {𝐴} ∧ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
3634, 35bitri 278 . . . . . 6 ((𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵) ↔ (𝑦 = {𝐴} ∧ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
3736exbii 1849 . . . . 5 (∃𝑦(𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵) ↔ ∃𝑦(𝑦 = {𝐴} ∧ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
38 breq1 5036 . . . . . . 7 (𝑦 = {𝐴} → (𝑦𝐹𝐵 {𝐴}𝐹𝐵))
3938anbi2d 631 . . . . . 6 (𝑦 = {𝐴} → ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵)))
4027, 39ceqsexv 3492 . . . . 5 (∃𝑦(𝑦 = {𝐴} ∧ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵))
4137, 40bitri 278 . . . 4 (∃𝑦(𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵))
4210, 41syl6bb 290 . . 3 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵)))
4342expcom 417 . 2 (𝐵𝑉 → (𝐴 ∈ V → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵))))
443, 6, 43pm5.21ndd 384 1 (𝐵𝑉 → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2112  Vcvv 3444   ∪ cun 3882  ∅c0 4246  {csn 4528  ∪ cuni 4803   class class class wbr 5033   ↦ cmpt 5113  ◡ccnv 5522  dom cdm 5523   ∘ ccom 5527  Fun wfun 6322  ‘cfv 6328  tpos ctpos 7878 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336  df-tpos 7879 This theorem is referenced by:  brtpos0  7886  reldmtpos  7887  brtpos  7888  dftpos4  7898  tpostpos  7899
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