Theorem brtpos2 8216

Description: Value of the transposition at an ordered 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 8215	. . . 4 ⊢ Rel tpos 𝐹
2	1	brrelex1i 5732	. . 3 ⊢ (𝐴tpos 𝐹𝐵 → 𝐴 ∈ V)
3	2	a1i 11	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴tpos 𝐹𝐵 → 𝐴 ∈ V))
4		elex 3492	. . . 4 ⊢ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) → 𝐴 ∈ V)
5	4	adantr 481	. . 3 ⊢ ((𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵) → 𝐴 ∈ V)
6	5	a1i 11	. 2 ⊢ (𝐵 ∈ 𝑉 → ((𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵) → 𝐴 ∈ V))
7		df-tpos 8210	. . . . . 6 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
8	7	breqi 5154	. . . . 5 ⊢ (𝐴tpos 𝐹𝐵 ↔ 𝐴(𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))𝐵)
9		brcog 5866	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴(𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))𝐵 ↔ ∃𝑦(𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ∧ 𝑦𝐹𝐵)))
10	8, 9	bitrid 282	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴tpos 𝐹𝐵 ↔ ∃𝑦(𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ∧ 𝑦𝐹𝐵)))
11		funmpt 6586	. . . . . . . . . . 11 ⊢ Fun (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
12		funbrfv2b 6949	. . . . . . . . . . 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		snex 5431	. . . . . . . . . . . . . . . 16 ⊢ {𝑥} ∈ V
15	14	cnvex 7915	. . . . . . . . . . . . . . 15 ⊢ ◡{𝑥} ∈ V
16	15	uniex 7730	. . . . . . . . . . . . . 14 ⊢ ∪ ◡{𝑥} ∈ V
17		eqid 2732	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
18	16, 17	dmmpti 6694	. . . . . . . . . . . . 13 ⊢ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = (◡dom 𝐹 ∪ {∅})
19	18	eleq2i 2825	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↔ 𝐴 ∈ (◡dom 𝐹 ∪ {∅}))
20		eqcom 2739	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴) = 𝑦 ↔ 𝑦 = ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴))
21	19, 20	anbi12i 627	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ∧ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴) = 𝑦) ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦 = ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴)))
22		sneq 4638	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
23	22	cnveqd 5875	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 → ◡{𝑥} = ◡{𝐴})
24	23	unieqd 4922	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐴 → ∪ ◡{𝑥} = ∪ ◡{𝐴})
25		snex 5431	. . . . . . . . . . . . . . . 16 ⊢ {𝐴} ∈ V
26	25	cnvex 7915	. . . . . . . . . . . . . . 15 ⊢ ◡{𝐴} ∈ V
27	26	uniex 7730	. . . . . . . . . . . . . 14 ⊢ ∪ ◡{𝐴} ∈ V
28	24, 17, 27	fvmpt 6998	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) → ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴) = ∪ ◡{𝐴})
29	28	eqeq2d 2743	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) → (𝑦 = ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴) ↔ 𝑦 = ∪ ◡{𝐴}))
30	29	pm5.32i 575	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦 = ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴)) ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦 = ∪ ◡{𝐴}))
31	21, 30	bitri 274	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ∧ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴) = 𝑦) ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦 = ∪ ◡{𝐴}))
32	13, 31	bitri 274	. . . . . . . . 9 ⊢ (𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦 = ∪ ◡{𝐴}))
33	32	biancomi 463	. . . . . . . 8 ⊢ (𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ↔ (𝑦 = ∪ ◡{𝐴} ∧ 𝐴 ∈ (◡dom 𝐹 ∪ {∅})))
34	33	anbi1i 624	. . . . . . 7 ⊢ ((𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ∧ 𝑦𝐹𝐵) ↔ ((𝑦 = ∪ ◡{𝐴} ∧ 𝐴 ∈ (◡dom 𝐹 ∪ {∅})) ∧ 𝑦𝐹𝐵))
35		anass 469	. . . . . . 7 ⊢ (((𝑦 = ∪ ◡{𝐴} ∧ 𝐴 ∈ (◡dom 𝐹 ∪ {∅})) ∧ 𝑦𝐹𝐵) ↔ (𝑦 = ∪ ◡{𝐴} ∧ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
36	34, 35	bitri 274	. . . . . 6 ⊢ ((𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ∧ 𝑦𝐹𝐵) ↔ (𝑦 = ∪ ◡{𝐴} ∧ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
37	36	exbii 1850	. . . . 5 ⊢ (∃𝑦(𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ∧ 𝑦𝐹𝐵) ↔ ∃𝑦(𝑦 = ∪ ◡{𝐴} ∧ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
38		breq1 5151	. . . . . . 7 ⊢ (𝑦 = ∪ ◡{𝐴} → (𝑦𝐹𝐵 ↔ ∪ ◡{𝐴}𝐹𝐵))
39	38	anbi2d 629	. . . . . 6 ⊢ (𝑦 = ∪ ◡{𝐴} → ((𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵) ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵)))
40	27, 39	ceqsexv 3525	. . . . 5 ⊢ (∃𝑦(𝑦 = ∪ ◡{𝐴} ∧ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)) ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵))
41	37, 40	bitri 274	. . . 4 ⊢ (∃𝑦(𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ∧ 𝑦𝐹𝐵) ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵))
42	10, 41	bitrdi 286	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵)))
43	42	expcom 414	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ V → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵))))
44	3, 6, 43	pm5.21ndd 380	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  Vcvv 3474   ∪ cun 3946  ∅c0 4322  {csn 4628  ∪ cuni 4908   class class class wbr 5148   ↦ cmpt 5231  ^◡ccnv 5675  dom cdm 5676   ∘ ccom 5680  Fun wfun 6537  ‘cfv 6543  tpos ctpos 8209

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551  df-tpos 8210

This theorem is referenced by:  brtpos0  8217  reldmtpos  8218  brtpos  8219  dftpos4  8229  tpostpos  8230