Theorem brtpos0 8235

Description: The behavior of tpos when the left argument is the empty set (which is not an ordered pair but is the "default" value of an ordered pair when the arguments are proper classes). This allows to eliminate sethood hypotheses on 𝐴, 𝐵 in brtpos 8237. (Contributed by Mario Carneiro, 10-Sep-2015.)

Assertion

Ref	Expression
brtpos0	⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴))

Proof of Theorem brtpos0

Step	Hyp	Ref	Expression
1		brtpos2 8234	. 2 ⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ (∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴)))
2		ssun2 4132	. . . . 5 ⊢ {∅} ⊆ (◡dom 𝐹 ∪ {∅})
3		0ex 5272	. . . . . 6 ⊢ ∅ ∈ V
4	3	snid 4630	. . . . 5 ⊢ ∅ ∈ {∅}
5	2, 4	sselii 3935	. . . 4 ⊢ ∅ ∈ (◡dom 𝐹 ∪ {∅})
6	5	biantrur 540	. . 3 ⊢ (∪ ◡{∅}𝐹𝐴 ↔ (∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴))
7		cnvsn0 6213	. . . . . 6 ⊢ ◡{∅} = ∅
8	7	unieqi 4886	. . . . 5 ⊢ ∪ ◡{∅} = ∪ ∅
9		uni0 4903	. . . . 5 ⊢ ∪ ∅ = ∅
10	8, 9	eqtri 2788	. . . 4 ⊢ ∪ ◡{∅} = ∅
11	10	breq1i 5118	. . 3 ⊢ (∪ ◡{∅}𝐹𝐴 ↔ ∅𝐹𝐴)
12	6, 11	bitr3i 280	. 2 ⊢ ((∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴) ↔ ∅𝐹𝐴)
13	1, 12	bitrdi 290	1 ⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2146   ∪ cun 3904  ∅c0 4286  {csn 4591  ∪ cuni 4874   class class class wbr 5111  ^◡ccnv 5662  dom cdm 5663  tpos ctpos 8227

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-fv 6548  df-tpos 8228

This theorem is used by:  reldmtpos  8236  brtpos  8237  tpostpos  8248  tposres0  49714