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Theorem brtpos2 8242
Description: Value of the transposition at an ordered 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 8241 . . . 4 Rel tpos 𝐹
21brrelex1i 5736 . . 3 (𝐴tpos 𝐹𝐵𝐴 ∈ V)
32a1i 11 . 2 (𝐵𝑉 → (𝐴tpos 𝐹𝐵𝐴 ∈ V))
4 elex 3490 . . . 4 (𝐴 ∈ (dom 𝐹 ∪ {∅}) → 𝐴 ∈ V)
54adantr 479 . . 3 ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵) → 𝐴 ∈ V)
65a1i 11 . 2 (𝐵𝑉 → ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵) → 𝐴 ∈ V))
7 df-tpos 8236 . . . . . 6 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
87breqi 5156 . . . . 5 (𝐴tpos 𝐹𝐵𝐴(𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))𝐵)
9 brcog 5871 . . . . 5 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴(𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))𝐵 ↔ ∃𝑦(𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵)))
108, 9bitrid 282 . . . 4 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴tpos 𝐹𝐵 ↔ ∃𝑦(𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵)))
11 funmpt 6594 . . . . . . . . . . 11 Fun (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})
12 funbrfv2b 6959 . . . . . . . . . . 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 5435 . . . . . . . . . . . . . . . 16 {𝑥} ∈ V
1514cnvex 7937 . . . . . . . . . . . . . . 15 {𝑥} ∈ V
1615uniex 7750 . . . . . . . . . . . . . 14 {𝑥} ∈ V
17 eqid 2727 . . . . . . . . . . . . . 14 (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) = (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})
1816, 17dmmpti 6702 . . . . . . . . . . . . 13 dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) = (dom 𝐹 ∪ {∅})
1918eleq2i 2820 . . . . . . . . . . . 12 (𝐴 ∈ dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ↔ 𝐴 ∈ (dom 𝐹 ∪ {∅}))
20 eqcom 2734 . . . . . . . . . . . 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 4640 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐴 → {𝑥} = {𝐴})
2322cnveqd 5880 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴{𝑥} = {𝐴})
2423unieqd 4923 . . . . . . . . . . . . . 14 (𝑥 = 𝐴 {𝑥} = {𝐴})
25 snex 5435 . . . . . . . . . . . . . . . 16 {𝐴} ∈ V
2625cnvex 7937 . . . . . . . . . . . . . . 15 {𝐴} ∈ V
2726uniex 7750 . . . . . . . . . . . . . 14 {𝐴} ∈ V
2824, 17, 27fvmpt 7008 . . . . . . . . . . . . 13 (𝐴 ∈ (dom 𝐹 ∪ {∅}) → ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) = {𝐴})
2928eqeq2d 2738 . . . . . . . . . . . 12 (𝐴 ∈ (dom 𝐹 ∪ {∅}) → (𝑦 = ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) ↔ 𝑦 = {𝐴}))
3029pm5.32i 573 . . . . . . . . . . 11 ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴)) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = {𝐴}))
3121, 30bitri 274 . . . . . . . . . 10 ((𝐴 ∈ dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ∧ ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) = 𝑦) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = {𝐴}))
3213, 31bitri 274 . . . . . . . . 9 (𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦 ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = {𝐴}))
3332biancomi 461 . . . . . . . 8 (𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦 ↔ (𝑦 = {𝐴} ∧ 𝐴 ∈ (dom 𝐹 ∪ {∅})))
3433anbi1i 622 . . . . . . 7 ((𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵) ↔ ((𝑦 = {𝐴} ∧ 𝐴 ∈ (dom 𝐹 ∪ {∅})) ∧ 𝑦𝐹𝐵))
35 anass 467 . . . . . . 7 (((𝑦 = {𝐴} ∧ 𝐴 ∈ (dom 𝐹 ∪ {∅})) ∧ 𝑦𝐹𝐵) ↔ (𝑦 = {𝐴} ∧ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
3634, 35bitri 274 . . . . . 6 ((𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵) ↔ (𝑦 = {𝐴} ∧ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
3736exbii 1842 . . . . 5 (∃𝑦(𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵) ↔ ∃𝑦(𝑦 = {𝐴} ∧ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
38 breq1 5153 . . . . . . 7 (𝑦 = {𝐴} → (𝑦𝐹𝐵 {𝐴}𝐹𝐵))
3938anbi2d 628 . . . . . 6 (𝑦 = {𝐴} → ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵)))
4027, 39ceqsexv 3523 . . . . 5 (∃𝑦(𝑦 = {𝐴} ∧ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵))
4137, 40bitri 274 . . . 4 (∃𝑦(𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵))
4210, 41bitrdi 286 . . 3 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵)))
4342expcom 412 . 2 (𝐵𝑉 → (𝐴 ∈ V → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵))))
443, 6, 43pm5.21ndd 378 1 (𝐵𝑉 → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wex 1773  wcel 2098  Vcvv 3471  cun 3945  c0 4324  {csn 4630   cuni 4910   class class class wbr 5150  cmpt 5233  ccnv 5679  dom cdm 5680  ccom 5684  Fun wfun 6545  cfv 6551  tpos ctpos 8235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-fv 6559  df-tpos 8236
This theorem is referenced by:  brtpos0  8243  reldmtpos  8244  brtpos  8245  dftpos4  8255  tpostpos  8256
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