Theorem brtpos2 6482

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 6481	. . . 4 ⊢ Rel tpos 𝐹
2	1	brrelex1i 4793	. . 3 ⊢ (𝐴tpos 𝐹𝐵 → 𝐴 ∈ V)
3	2	a1i 9	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴tpos 𝐹𝐵 → 𝐴 ∈ V))
4		elex 2825	. . . 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 6476	. . . . . 6 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
8	7	breqi 4115	. . . . 5 ⊢ (𝐴tpos 𝐹𝐵 ↔ 𝐴(𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))𝐵)
9		brcog 4922	. . . . 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 5390	. . . . . . . . . . 11 ⊢ Fun (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
12		funbrfv2b 5721	. . . . . . . . . . 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 2816	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
15		snexg 4297	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ V → {𝑥} ∈ V)
16	14, 15	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ {𝑥} ∈ V
17	16	cnvex 5301	. . . . . . . . . . . . . . 15 ⊢ ◡{𝑥} ∈ V
18	17	uniex 4558	. . . . . . . . . . . . . 14 ⊢ ∪ ◡{𝑥} ∈ V
19		eqid 2232	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
20	18, 19	dmmpti 5488	. . . . . . . . . . . . 13 ⊢ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = (◡dom 𝐹 ∪ {∅})
21	20	eleq2i 2299	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↔ 𝐴 ∈ (◡dom 𝐹 ∪ {∅}))
22		eqcom 2234	. . . . . . . . . . . 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 4297	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) → {𝐴} ∈ V)
25		cnvexg 5300	. . . . . . . . . . . . . . . 16 ⊢ ({𝐴} ∈ V → ◡{𝐴} ∈ V)
26	24, 25	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) → ◡{𝐴} ∈ V)
27		uniexg 4560	. . . . . . . . . . . . . . 15 ⊢ (◡{𝐴} ∈ V → ∪ ◡{𝐴} ∈ V)
28	26, 27	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) → ∪ ◡{𝐴} ∈ V)
29		sneq 3700	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
30	29	cnveqd 4931	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐴 → ◡{𝑥} = ◡{𝐴})
31	30	unieqd 3925	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 → ∪ ◡{𝑥} = ∪ ◡{𝐴})
32	31, 19	fvmptg 5753	. . . . . . . . . . . . . 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 2244	. . . . . . . . . . . 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 1654	. . . . 5 ⊢ (∃𝑦(𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ∧ 𝑦𝐹𝐵) ↔ ∃𝑦(𝑦 = ∪ ◡{𝐴} ∧ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
44		exsimpr 1667	. . . . . . 7 ⊢ (∃𝑦(𝑦 = ∪ ◡{𝐴} ∧ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)) → ∃𝑦(𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵))
45		exsimpl 1666	. . . . . . . 8 ⊢ (∃𝑦(𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵) → ∃𝑦 𝐴 ∈ (◡dom 𝐹 ∪ {∅}))
46		19.9v 1920	. . . . . . . 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 4112	. . . . . . . . 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 2946	. . . . . . 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 712	. . . . 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 713	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2203  Vcvv 2813   ∪ cun 3209  ∅c0 3508  {csn 3689  ∪ cuni 3914   class class class wbr 4109   ↦ cmpt 4171  ^◡ccnv 4748  dom cdm 4749   ∘ ccom 4753  Fun wfun 5346  ‘cfv 5352  tpos ctpos 6475

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-tpos 6476

This theorem is referenced by:  brtpos0  6483  reldmtpos  6484  brtposg  6485  dftpos4  6494  tpostpos  6495