Theorem brtpos0 8212

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 8214. (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 8211	. 2 ⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ (∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴)))
2		ssun2 4142	. . . . 5 ⊢ {∅} ⊆ (◡dom 𝐹 ∪ {∅})
3		0ex 5262	. . . . . 6 ⊢ ∅ ∈ V
4	3	snid 4626	. . . . 5 ⊢ ∅ ∈ {∅}
5	2, 4	sselii 3943	. . . 4 ⊢ ∅ ∈ (◡dom 𝐹 ∪ {∅})
6	5	biantrur 530	. . 3 ⊢ (∪ ◡{∅}𝐹𝐴 ↔ (∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴))
7		cnvsn0 6183	. . . . . 6 ⊢ ◡{∅} = ∅
8	7	unieqi 4883	. . . . 5 ⊢ ∪ ◡{∅} = ∪ ∅
9		uni0 4899	. . . . 5 ⊢ ∪ ∅ = ∅
10	8, 9	eqtri 2752	. . . 4 ⊢ ∪ ◡{∅} = ∅
11	10	breq1i 5114	. . 3 ⊢ (∪ ◡{∅}𝐹𝐴 ↔ ∅𝐹𝐴)
12	6, 11	bitr3i 277	. 2 ⊢ ((∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴) ↔ ∅𝐹𝐴)
13	1, 12	bitrdi 287	1 ⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2109   ∪ cun 3912  ∅c0 4296  {csn 4589  ∪ cuni 4871   class class class wbr 5107  ^◡ccnv 5637  dom cdm 5638  tpos ctpos 8204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-fv 6519  df-tpos 8205

This theorem is referenced by:  reldmtpos  8213  brtpos  8214  tpostpos  8225  tposres0  48865