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Theorem brtpos2 6245
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 6244 . . . 4 Rel tpos 𝐹
21brrelex1i 4665 . . 3 (𝐴tpos 𝐹𝐵𝐴 ∈ V)
32a1i 9 . 2 (𝐵𝑉 → (𝐴tpos 𝐹𝐵𝐴 ∈ V))
4 elex 2748 . . . 4 (𝐴 ∈ (dom 𝐹 ∪ {∅}) → 𝐴 ∈ V)
54adantr 276 . . 3 ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵) → 𝐴 ∈ V)
65a1i 9 . 2 (𝐵𝑉 → ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵) → 𝐴 ∈ V))
7 df-tpos 6239 . . . . . 6 tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))
87breqi 4006 . . . . 5 (𝐴tpos 𝐹𝐵𝐴(𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))𝐵)
9 brcog 4789 . . . . 5 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴(𝐹 ∘ (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}))𝐵 ↔ ∃𝑦(𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵)))
108, 9bitrid 192 . . . 4 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴tpos 𝐹𝐵 ↔ ∃𝑦(𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵)))
11 funmpt 5249 . . . . . . . . . . 11 Fun (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})
12 funbrfv2b 5555 . . . . . . . . . . 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 vex 2740 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
15 snexg 4181 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ V → {𝑥} ∈ V)
1614, 15ax-mp 5 . . . . . . . . . . . . . . . 16 {𝑥} ∈ V
1716cnvex 5162 . . . . . . . . . . . . . . 15 {𝑥} ∈ V
1817uniex 4433 . . . . . . . . . . . . . 14 {𝑥} ∈ V
19 eqid 2177 . . . . . . . . . . . . . 14 (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) = (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})
2018, 19dmmpti 5340 . . . . . . . . . . . . 13 dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) = (dom 𝐹 ∪ {∅})
2120eleq2i 2244 . . . . . . . . . . . 12 (𝐴 ∈ dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ↔ 𝐴 ∈ (dom 𝐹 ∪ {∅}))
22 eqcom 2179 . . . . . . . . . . . 12 (((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) = 𝑦𝑦 = ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴))
2321, 22anbi12i 460 . . . . . . . . . . 11 ((𝐴 ∈ dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ∧ ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) = 𝑦) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴)))
24 snexg 4181 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (dom 𝐹 ∪ {∅}) → {𝐴} ∈ V)
25 cnvexg 5161 . . . . . . . . . . . . . . . 16 ({𝐴} ∈ V → {𝐴} ∈ V)
2624, 25syl 14 . . . . . . . . . . . . . . 15 (𝐴 ∈ (dom 𝐹 ∪ {∅}) → {𝐴} ∈ V)
27 uniexg 4435 . . . . . . . . . . . . . . 15 ({𝐴} ∈ V → {𝐴} ∈ V)
2826, 27syl 14 . . . . . . . . . . . . . 14 (𝐴 ∈ (dom 𝐹 ∪ {∅}) → {𝐴} ∈ V)
29 sneq 3602 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝐴 → {𝑥} = {𝐴})
3029cnveqd 4798 . . . . . . . . . . . . . . . 16 (𝑥 = 𝐴{𝑥} = {𝐴})
3130unieqd 3818 . . . . . . . . . . . . . . 15 (𝑥 = 𝐴 {𝑥} = {𝐴})
3231, 19fvmptg 5587 . . . . . . . . . . . . . 14 ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴} ∈ V) → ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) = {𝐴})
3328, 32mpdan 421 . . . . . . . . . . . . 13 (𝐴 ∈ (dom 𝐹 ∪ {∅}) → ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) = {𝐴})
3433eqeq2d 2189 . . . . . . . . . . . 12 (𝐴 ∈ (dom 𝐹 ∪ {∅}) → (𝑦 = ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) ↔ 𝑦 = {𝐴}))
3534pm5.32i 454 . . . . . . . . . . 11 ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴)) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = {𝐴}))
3623, 35bitri 184 . . . . . . . . . 10 ((𝐴 ∈ dom (𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥}) ∧ ((𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})‘𝐴) = 𝑦) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = {𝐴}))
3713, 36bitri 184 . . . . . . . . 9 (𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦 ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = {𝐴}))
38 ancom 266 . . . . . . . . 9 ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦 = {𝐴}) ↔ (𝑦 = {𝐴} ∧ 𝐴 ∈ (dom 𝐹 ∪ {∅})))
3937, 38bitri 184 . . . . . . . 8 (𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦 ↔ (𝑦 = {𝐴} ∧ 𝐴 ∈ (dom 𝐹 ∪ {∅})))
4039anbi1i 458 . . . . . . 7 ((𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵) ↔ ((𝑦 = {𝐴} ∧ 𝐴 ∈ (dom 𝐹 ∪ {∅})) ∧ 𝑦𝐹𝐵))
41 anass 401 . . . . . . 7 (((𝑦 = {𝐴} ∧ 𝐴 ∈ (dom 𝐹 ∪ {∅})) ∧ 𝑦𝐹𝐵) ↔ (𝑦 = {𝐴} ∧ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
4240, 41bitri 184 . . . . . 6 ((𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵) ↔ (𝑦 = {𝐴} ∧ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
4342exbii 1605 . . . . 5 (∃𝑦(𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵) ↔ ∃𝑦(𝑦 = {𝐴} ∧ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
44 exsimpr 1618 . . . . . . 7 (∃𝑦(𝑦 = {𝐴} ∧ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)) → ∃𝑦(𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵))
45 exsimpl 1617 . . . . . . . 8 (∃𝑦(𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵) → ∃𝑦 𝐴 ∈ (dom 𝐹 ∪ {∅}))
46 19.9v 1871 . . . . . . . 8 (∃𝑦 𝐴 ∈ (dom 𝐹 ∪ {∅}) ↔ 𝐴 ∈ (dom 𝐹 ∪ {∅}))
4745, 46sylib 122 . . . . . . 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 109 . . . . . 6 ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵) → 𝐴 ∈ (dom 𝐹 ∪ {∅}))
50 breq1 4003 . . . . . . . . 9 (𝑦 = {𝐴} → (𝑦𝐹𝐵 {𝐴}𝐹𝐵))
5150anbi2d 464 . . . . . . . 8 (𝑦 = {𝐴} → ((𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵)))
5251ceqsexgv 2866 . . . . . . 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 704 . . . . 5 (∃𝑦(𝑦 = {𝐴} ∧ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵))
5543, 54bitri 184 . . . 4 (∃𝑦(𝐴(𝑥 ∈ (dom 𝐹 ∪ {∅}) ↦ {𝑥})𝑦𝑦𝐹𝐵) ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵))
5610, 55bitrdi 196 . . 3 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵)))
5756expcom 116 . 2 (𝐵𝑉 → (𝐴 ∈ V → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵))))
583, 6, 57pm5.21ndd 705 1 (𝐵𝑉 → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (dom 𝐹 ∪ {∅}) ∧ {𝐴}𝐹𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wex 1492  wcel 2148  Vcvv 2737  cun 3127  c0 3422  {csn 3591   cuni 3807   class class class wbr 4000  cmpt 4061  ccnv 4621  dom cdm 4622  ccom 4626  Fun wfun 5205  cfv 5211  tpos ctpos 6238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-fv 5219  df-tpos 6239
This theorem is referenced by:  brtpos0  6246  reldmtpos  6247  brtposg  6248  dftpos4  6257  tpostpos  6258
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