Theorem brtpos0 8250

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 8252. (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 8249	. 2 ⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ (∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴)))
2		ssun2 4125	. . . . 5 ⊢ {∅} ⊆ (◡dom 𝐹 ∪ {∅})
3		0ex 5261	. . . . . 6 ⊢ ∅ ∈ V
4	3	snid 4623	. . . . 5 ⊢ ∅ ∈ {∅}
5	2, 4	sselii 3928	. . . 4 ⊢ ∅ ∈ (◡dom 𝐹 ∪ {∅})
6	5	biantrur 540	. . 3 ⊢ (∪ ◡{∅}𝐹𝐴 ↔ (∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴))
7		cnvsn0 6211	. . . . . 6 ⊢ ◡{∅} = ∅
8	7	unieqi 4879	. . . . 5 ⊢ ∪ ◡{∅} = ∪ ∅
9		uni0 4896	. . . . 5 ⊢ ∪ ∅ = ∅
10	8, 9	eqtri 2784	. . . 4 ⊢ ∪ ◡{∅} = ∅
11	10	breq1i 5110	. . 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 2145   ∪ cun 3897  ∅c0 4279  {csn 4584  ∪ cuni 4867   class class class wbr 5103  ^◡ccnv 5650  dom cdm 5651  tpos ctpos 8242

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546  df-tpos 8243

This theorem is used by:  reldmtpos  8251  brtpos  8252  tpostpos  8263  tposres0  49984