Theorem brtpos0 8217

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 8219. (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 8216	. 2 ⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ (∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴)))
2		ssun2 4173	. . . . 5 ⊢ {∅} ⊆ (◡dom 𝐹 ∪ {∅})
3		0ex 5307	. . . . . 6 ⊢ ∅ ∈ V
4	3	snid 4664	. . . . 5 ⊢ ∅ ∈ {∅}
5	2, 4	sselii 3979	. . . 4 ⊢ ∅ ∈ (◡dom 𝐹 ∪ {∅})
6	5	biantrur 531	. . 3 ⊢ (∪ ◡{∅}𝐹𝐴 ↔ (∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴))
7		cnvsn0 6209	. . . . . 6 ⊢ ◡{∅} = ∅
8	7	unieqi 4921	. . . . 5 ⊢ ∪ ◡{∅} = ∪ ∅
9		uni0 4939	. . . . 5 ⊢ ∪ ∅ = ∅
10	8, 9	eqtri 2760	. . . 4 ⊢ ∪ ◡{∅} = ∅
11	10	breq1i 5155	. . 3 ⊢ (∪ ◡{∅}𝐹𝐴 ↔ ∅𝐹𝐴)
12	6, 11	bitr3i 276	. 2 ⊢ ((∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴) ↔ ∅𝐹𝐴)
13	1, 12	bitrdi 286	1 ⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106   ∪ cun 3946  ∅c0 4322  {csn 4628  ∪ cuni 4908   class class class wbr 5148  ^◡ccnv 5675  dom cdm 5676  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:  reldmtpos  8218  brtpos  8219  tpostpos  8230