Theorem brtpos2 6337

Description: Value of the transposition at a pair ⟨𝐴, 𝐵⟩. (Contributed by Mario Carneiro, 10-Sep-2015.)

Assertion

Ref	Expression
brtpos2	⊢ (𝐵 ∈ 𝑉 → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵)))

Proof of Theorem brtpos2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reltpos 6336	. . . 4 ⊢ Rel tpos 𝐹
2	1	brrelex1i 4718	. . 3 ⊢ (𝐴tpos 𝐹𝐵 → 𝐴 ∈ V)
3	2	a1i 9	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴tpos 𝐹𝐵 → 𝐴 ∈ V))
4		elex 2783	. . . 4 ⊢ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) → 𝐴 ∈ V)
5	4	adantr 276	. . 3 ⊢ ((𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵) → 𝐴 ∈ V)
6	5	a1i 9	. 2 ⊢ (𝐵 ∈ 𝑉 → ((𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵) → 𝐴 ∈ V))
7		df-tpos 6331	. . . . . 6 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
8	7	breqi 4050	. . . . 5 ⊢ (𝐴tpos 𝐹𝐵 ↔ 𝐴(𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))𝐵)
9		brcog 4845	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴(𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))𝐵 ↔ ∃𝑦(𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ∧ 𝑦𝐹𝐵)))
10	8, 9	bitrid 192	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴tpos 𝐹𝐵 ↔ ∃𝑦(𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ∧ 𝑦𝐹𝐵)))
11		funmpt 5309	. . . . . . . . . . 11 ⊢ Fun (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
12		funbrfv2b 5623	. . . . . . . . . . 11 ⊢ (Fun (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) → (𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ↔ (𝐴 ∈ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ∧ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴) = 𝑦)))
13	11, 12	ax-mp 5	. . . . . . . . . 10 ⊢ (𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ↔ (𝐴 ∈ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ∧ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴) = 𝑦))
14		vex 2775	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
15		snexg 4228	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ V → {𝑥} ∈ V)
16	14, 15	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ {𝑥} ∈ V
17	16	cnvex 5221	. . . . . . . . . . . . . . 15 ⊢ ◡{𝑥} ∈ V
18	17	uniex 4484	. . . . . . . . . . . . . 14 ⊢ ∪ ◡{𝑥} ∈ V
19		eqid 2205	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
20	18, 19	dmmpti 5405	. . . . . . . . . . . . 13 ⊢ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = (◡dom 𝐹 ∪ {∅})
21	20	eleq2i 2272	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↔ 𝐴 ∈ (◡dom 𝐹 ∪ {∅}))
22		eqcom 2207	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴) = 𝑦 ↔ 𝑦 = ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴))
23	21, 22	anbi12i 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 4228	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) → {𝐴} ∈ V)
25		cnvexg 5220	. . . . . . . . . . . . . . . 16 ⊢ ({𝐴} ∈ V → ◡{𝐴} ∈ V)
26	24, 25	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) → ◡{𝐴} ∈ V)
27		uniexg 4486	. . . . . . . . . . . . . . 15 ⊢ (◡{𝐴} ∈ V → ∪ ◡{𝐴} ∈ V)
28	26, 27	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) → ∪ ◡{𝐴} ∈ V)
29		sneq 3644	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
30	29	cnveqd 4854	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐴 → ◡{𝑥} = ◡{𝐴})
31	30	unieqd 3861	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 → ∪ ◡{𝑥} = ∪ ◡{𝐴})
32	31, 19	fvmptg 5655	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴} ∈ V) → ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴) = ∪ ◡{𝐴})
33	28, 32	mpdan 421	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) → ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴) = ∪ ◡{𝐴})
34	33	eqeq2d 2217	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) → (𝑦 = ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴) ↔ 𝑦 = ∪ ◡{𝐴}))
35	34	pm5.32i 454	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦 = ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴)) ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦 = ∪ ◡{𝐴}))
36	23, 35	bitri 184	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ∧ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴) = 𝑦) ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦 = ∪ ◡{𝐴}))
37	13, 36	bitri 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 ∪ {∅})))
39	37, 38	bitri 184	. . . . . . . 8 ⊢ (𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ↔ (𝑦 = ∪ ◡{𝐴} ∧ 𝐴 ∈ (◡dom 𝐹 ∪ {∅})))
40	39	anbi1i 458	. . . . . . 7 ⊢ ((𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ∧ 𝑦𝐹𝐵) ↔ ((𝑦 = ∪ ◡{𝐴} ∧ 𝐴 ∈ (◡dom 𝐹 ∪ {∅})) ∧ 𝑦𝐹𝐵))
41		anass 401	. . . . . . 7 ⊢ (((𝑦 = ∪ ◡{𝐴} ∧ 𝐴 ∈ (◡dom 𝐹 ∪ {∅})) ∧ 𝑦𝐹𝐵) ↔ (𝑦 = ∪ ◡{𝐴} ∧ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
42	40, 41	bitri 184	. . . . . 6 ⊢ ((𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ∧ 𝑦𝐹𝐵) ↔ (𝑦 = ∪ ◡{𝐴} ∧ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
43	42	exbii 1628	. . . . 5 ⊢ (∃𝑦(𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ∧ 𝑦𝐹𝐵) ↔ ∃𝑦(𝑦 = ∪ ◡{𝐴} ∧ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
44		exsimpr 1641	. . . . . . 7 ⊢ (∃𝑦(𝑦 = ∪ ◡{𝐴} ∧ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)) → ∃𝑦(𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵))
45		exsimpl 1640	. . . . . . . 8 ⊢ (∃𝑦(𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵) → ∃𝑦 𝐴 ∈ (◡dom 𝐹 ∪ {∅}))
46		19.9v 1894	. . . . . . . 8 ⊢ (∃𝑦 𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ↔ 𝐴 ∈ (◡dom 𝐹 ∪ {∅}))
47	45, 46	sylib 122	. . . . . . 7 ⊢ (∃𝑦(𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵) → 𝐴 ∈ (◡dom 𝐹 ∪ {∅}))
48	44, 47	syl 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 4047	. . . . . . . . 9 ⊢ (𝑦 = ∪ ◡{𝐴} → (𝑦𝐹𝐵 ↔ ∪ ◡{𝐴}𝐹𝐵))
51	50	anbi2d 464	. . . . . . . 8 ⊢ (𝑦 = ∪ ◡{𝐴} → ((𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵) ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵)))
52	51	ceqsexgv 2902	. . . . . . 7 ⊢ (∪ ◡{𝐴} ∈ V → (∃𝑦(𝑦 = ∪ ◡{𝐴} ∧ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)) ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵)))
53	28, 52	syl 14	. . . . . 6 ⊢ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) → (∃𝑦(𝑦 = ∪ ◡{𝐴} ∧ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)) ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵)))
54	48, 49, 53	pm5.21nii 706	. . . . 5 ⊢ (∃𝑦(𝑦 = ∪ ◡{𝐴} ∧ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)) ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵))
55	43, 54	bitri 184	. . . 4 ⊢ (∃𝑦(𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ∧ 𝑦𝐹𝐵) ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵))
56	10, 55	bitrdi 196	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵)))
57	56	expcom 116	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ V → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵))))
58	3, 6, 57	pm5.21ndd 707	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373  ∃wex 1515   ∈ wcel 2176  Vcvv 2772   ∪ cun 3164  ∅c0 3460  {csn 3633  ∪ cuni 3850   class class class wbr 4044   ↦ cmpt 4105  ^◡ccnv 4674  dom cdm 4675   ∘ ccom 4679  Fun wfun 5265  ‘cfv 5271  tpos ctpos 6330

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-tpos 6331

This theorem is referenced by:  brtpos0  6338  reldmtpos  6339  brtposg  6340  dftpos4  6349  tpostpos  6350