Theorem brtpos2 8224

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 8223	. . . 4 ⊢ Rel tpos 𝐹
2	1	brrelex1i 5717	. . 3 ⊢ (𝐴tpos 𝐹𝐵 → 𝐴 ∈ V)
3	2	a1i 11	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴tpos 𝐹𝐵 → 𝐴 ∈ V))
4		elex 3476	. . . 4 ⊢ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) → 𝐴 ∈ V)
5	4	adantr 485	. . 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 8218	. . . . . 6 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
8	7	breqi 5115	. . . . 5 ⊢ (𝐴tpos 𝐹𝐵 ↔ 𝐴(𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))𝐵)
9		brcog 5852	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴(𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))𝐵 ↔ ∃𝑦(𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ∧ 𝑦𝐹𝐵)))
10	8, 9	bitrid 286	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴tpos 𝐹𝐵 ↔ ∃𝑦(𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ∧ 𝑦𝐹𝐵)))
11		funmpt 6574	. . . . . . . . . . 11 ⊢ Fun (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
12		funbrfv2b 6938	. . . . . . . . . . 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 5410	. . . . . . . . . . . . . . . 16 ⊢ {𝑥} ∈ V
15	14	cnvex 7918	. . . . . . . . . . . . . . 15 ⊢ ◡{𝑥} ∈ V
16	15	uniex 7739	. . . . . . . . . . . . . 14 ⊢ ∪ ◡{𝑥} ∈ V
17		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
18	16, 17	dmmpti 6679	. . . . . . . . . . . . 13 ⊢ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = (◡dom 𝐹 ∪ {∅})
19	18	eleq2i 2855	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↔ 𝐴 ∈ (◡dom 𝐹 ∪ {∅}))
20		eqcom 2770	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴) = 𝑦 ↔ 𝑦 = ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴))
21	19, 20	anbi12i 639	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ∧ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴) = 𝑦) ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦 = ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴)))
22		sneq 4599	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
23	22	cnveqd 5861	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 → ◡{𝑥} = ◡{𝐴})
24	23	unieqd 4885	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐴 → ∪ ◡{𝑥} = ∪ ◡{𝐴})
25		snex 5410	. . . . . . . . . . . . . . . 16 ⊢ {𝐴} ∈ V
26	25	cnvex 7918	. . . . . . . . . . . . . . 15 ⊢ ◡{𝐴} ∈ V
27	26	uniex 7739	. . . . . . . . . . . . . 14 ⊢ ∪ ◡{𝐴} ∈ V
28	24, 17, 27	fvmpt 6989	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) → ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴) = ∪ ◡{𝐴})
29	28	eqeq2d 2774	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) → (𝑦 = ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴) ↔ 𝑦 = ∪ ◡{𝐴}))
30	29	pm5.32i 584	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦 = ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴)) ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦 = ∪ ◡{𝐴}))
31	21, 30	bitri 278	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ∧ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴) = 𝑦) ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦 = ∪ ◡{𝐴}))
32	13, 31	bitri 278	. . . . . . . . 9 ⊢ (𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦 = ∪ ◡{𝐴}))
33	32	biancomi 467	. . . . . . . 8 ⊢ (𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ↔ (𝑦 = ∪ ◡{𝐴} ∧ 𝐴 ∈ (◡dom 𝐹 ∪ {∅})))
34	33	anbi1i 635	. . . . . . 7 ⊢ ((𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ∧ 𝑦𝐹𝐵) ↔ ((𝑦 = ∪ ◡{𝐴} ∧ 𝐴 ∈ (◡dom 𝐹 ∪ {∅})) ∧ 𝑦𝐹𝐵))
35		anass 473	. . . . . . 7 ⊢ (((𝑦 = ∪ ◡{𝐴} ∧ 𝐴 ∈ (◡dom 𝐹 ∪ {∅})) ∧ 𝑦𝐹𝐵) ↔ (𝑦 = ∪ ◡{𝐴} ∧ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
36	34, 35	bitri 278	. . . . . 6 ⊢ ((𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ∧ 𝑦𝐹𝐵) ↔ (𝑦 = ∪ ◡{𝐴} ∧ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
37	36	exbii 1878	. . . . 5 ⊢ (∃𝑦(𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ∧ 𝑦𝐹𝐵) ↔ ∃𝑦(𝑦 = ∪ ◡{𝐴} ∧ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
38		breq1 5112	. . . . . . 7 ⊢ (𝑦 = ∪ ◡{𝐴} → (𝑦𝐹𝐵 ↔ ∪ ◡{𝐴}𝐹𝐵))
39	38	anbi2d 641	. . . . . 6 ⊢ (𝑦 = ∪ ◡{𝐴} → ((𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵) ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵)))
40	27, 39	ceqsexv 3503	. . . . 5 ⊢ (∃𝑦(𝑦 = ∪ ◡{𝐴} ∧ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)) ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵))
41	37, 40	bitri 278	. . . 4 ⊢ (∃𝑦(𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ∧ 𝑦𝐹𝐵) ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵))
42	10, 41	bitrdi 290	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵)))
43	42	expcom 418	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ V → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵))))
44	3, 6, 43	pm5.21ndd 382	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  Vcvv 3455   ∪ cun 3903  ∅c0 4286  {csn 4589  ∪ cuni 4872   class class class wbr 5109   ↦ cmpt 5192  ^◡ccnv 5660  dom cdm 5661   ∘ ccom 5665  Fun wfun 6530  ‘cfv 6536  tpos ctpos 8217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544  df-tpos 8218

This theorem is referenced by:  brtpos0  8225  reldmtpos  8226  brtpos  8227  dftpos4  8237  tpostpos  8238