Theorem brtpos2 7245

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 7244	. . . 4 ⊢ Rel tpos 𝐹
2	1	brrelexi 5082	. . 3 ⊢ (𝐴tpos 𝐹𝐵 → 𝐴 ∈ V)
3	2	a1i 11	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴tpos 𝐹𝐵 → 𝐴 ∈ V))
4		elex 3185	. . . 4 ⊢ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) → 𝐴 ∈ V)
5	4	adantr 480	. . 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 7239	. . . . . 6 ⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))
8	7	breqi 4589	. . . . 5 ⊢ (𝐴tpos 𝐹𝐵 ↔ 𝐴(𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))𝐵)
9		brcog 5210	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴(𝐹 ∘ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}))𝐵 ↔ ∃𝑦(𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ∧ 𝑦𝐹𝐵)))
10	8, 9	syl5bb 271	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴tpos 𝐹𝐵 ↔ ∃𝑦(𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ∧ 𝑦𝐹𝐵)))
11		funmpt 5840	. . . . . . . . . . 11 ⊢ Fun (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
12		funbrfv2b 6150	. . . . . . . . . . 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 4835	. . . . . . . . . . . . . . . 16 ⊢ {𝑥} ∈ V
15	14	cnvex 7006	. . . . . . . . . . . . . . 15 ⊢ ◡{𝑥} ∈ V
16	15	uniex 6851	. . . . . . . . . . . . . 14 ⊢ ∪ ◡{𝑥} ∈ V
17		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})
18	16, 17	dmmpti 5936	. . . . . . . . . . . . 13 ⊢ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) = (◡dom 𝐹 ∪ {∅})
19	18	eleq2i 2680	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ↔ 𝐴 ∈ (◡dom 𝐹 ∪ {∅}))
20		eqcom 2617	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴) = 𝑦 ↔ 𝑦 = ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴))
21	19, 20	anbi12i 729	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ∧ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴) = 𝑦) ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦 = ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴)))
22		sneq 4135	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
23	22	cnveqd 5220	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 → ◡{𝑥} = ◡{𝐴})
24	23	unieqd 4382	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝐴 → ∪ ◡{𝑥} = ∪ ◡{𝐴})
25		snex 4835	. . . . . . . . . . . . . . . 16 ⊢ {𝐴} ∈ V
26	25	cnvex 7006	. . . . . . . . . . . . . . 15 ⊢ ◡{𝐴} ∈ V
27	26	uniex 6851	. . . . . . . . . . . . . 14 ⊢ ∪ ◡{𝐴} ∈ V
28	24, 17, 27	fvmpt 6191	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) → ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴) = ∪ ◡{𝐴})
29	28	eqeq2d 2620	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) → (𝑦 = ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴) ↔ 𝑦 = ∪ ◡{𝐴}))
30	29	pm5.32i 667	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦 = ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴)) ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦 = ∪ ◡{𝐴}))
31	21, 30	bitri 263	. . . . . . . . . 10 ⊢ ((𝐴 ∈ dom (𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥}) ∧ ((𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})‘𝐴) = 𝑦) ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦 = ∪ ◡{𝐴}))
32	13, 31	bitri 263	. . . . . . . . 9 ⊢ (𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦 = ∪ ◡{𝐴}))
33		ancom 465	. . . . . . . . 9 ⊢ ((𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦 = ∪ ◡{𝐴}) ↔ (𝑦 = ∪ ◡{𝐴} ∧ 𝐴 ∈ (◡dom 𝐹 ∪ {∅})))
34	32, 33	bitri 263	. . . . . . . 8 ⊢ (𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ↔ (𝑦 = ∪ ◡{𝐴} ∧ 𝐴 ∈ (◡dom 𝐹 ∪ {∅})))
35	34	anbi1i 727	. . . . . . 7 ⊢ ((𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ∧ 𝑦𝐹𝐵) ↔ ((𝑦 = ∪ ◡{𝐴} ∧ 𝐴 ∈ (◡dom 𝐹 ∪ {∅})) ∧ 𝑦𝐹𝐵))
36		anass 679	. . . . . . 7 ⊢ (((𝑦 = ∪ ◡{𝐴} ∧ 𝐴 ∈ (◡dom 𝐹 ∪ {∅})) ∧ 𝑦𝐹𝐵) ↔ (𝑦 = ∪ ◡{𝐴} ∧ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
37	35, 36	bitri 263	. . . . . 6 ⊢ ((𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ∧ 𝑦𝐹𝐵) ↔ (𝑦 = ∪ ◡{𝐴} ∧ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
38	37	exbii 1764	. . . . 5 ⊢ (∃𝑦(𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ∧ 𝑦𝐹𝐵) ↔ ∃𝑦(𝑦 = ∪ ◡{𝐴} ∧ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)))
39		breq1 4586	. . . . . . 7 ⊢ (𝑦 = ∪ ◡{𝐴} → (𝑦𝐹𝐵 ↔ ∪ ◡{𝐴}𝐹𝐵))
40	39	anbi2d 736	. . . . . 6 ⊢ (𝑦 = ∪ ◡{𝐴} → ((𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵) ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵)))
41	27, 40	ceqsexv 3215	. . . . 5 ⊢ (∃𝑦(𝑦 = ∪ ◡{𝐴} ∧ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ 𝑦𝐹𝐵)) ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵))
42	38, 41	bitri 263	. . . 4 ⊢ (∃𝑦(𝐴(𝑥 ∈ (◡dom 𝐹 ∪ {∅}) ↦ ∪ ◡{𝑥})𝑦 ∧ 𝑦𝐹𝐵) ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵))
43	10, 42	syl6bb 275	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝑉) → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵)))
44	43	expcom 450	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐴 ∈ V → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵))))
45	3, 6, 44	pm5.21ndd 368	1 ⊢ (𝐵 ∈ 𝑉 → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{𝐴}𝐹𝐵)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538  ∅c0 3874  {csn 4125  ∪ cuni 4372   class class class wbr 4583   ↦ cmpt 4643  ^◡ccnv 5037  dom cdm 5038   ∘ ccom 5042  Fun wfun 5798  ‘cfv 5804  tpos ctpos 7238

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-tpos 7239

This theorem is referenced by:  brtpos0  7246  reldmtpos  7247  brtpos  7248  dftpos4  7258  tpostpos  7259