Theorem brtpos0 8232

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 8234. (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 8231	. 2 ⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ (∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴)))
2		ssun2 4125	. . . . 5 ⊢ {∅} ⊆ (◡dom 𝐹 ∪ {∅})
3		0ex 5264	. . . . . 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 6206	. . . . . 6 ⊢ ◡{∅} = ∅
8	7	unieqi 4879	. . . . 5 ⊢ ∪ ◡{∅} = ∪ ∅
9		uni0 4896	. . . . 5 ⊢ ∪ ∅ = ∅
10	8, 9	eqtri 2783	. . . 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 5654  dom cdm 5655  tpos ctpos 8224

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 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  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 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-fv 6541  df-tpos 8225

This theorem is used by:  reldmtpos  8233  brtpos  8234  tpostpos  8245  tposres0  49804