MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tz7.44lem1 Structured version   Visualization version   GIF version

Theorem tz7.44lem1 8406
Description: The ordered pair abstraction 𝐺 defined in the hypothesis is a function. This was a lemma for tz7.44-1 8407, tz7.44-2 8408, and tz7.44-3 8409 when they used that definition of 𝐺. Now, they use the maps-to df-mpt 5225 idiom so this lemma is not needed anymore, but is kept in case other applications (for instance in intuitionistic set theory) need it. (Contributed by NM, 23-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.)
Hypothesis
Ref Expression
tz7.44lem1.1 𝐺 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥 dom 𝑥))) ∨ (Lim dom 𝑥𝑦 = ran 𝑥))}
Assertion
Ref Expression
tz7.44lem1 Fun 𝐺
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑦,𝐻
Allowed substitution hints:   𝐴(𝑥)   𝐺(𝑥,𝑦)   𝐻(𝑥)

Proof of Theorem tz7.44lem1
StepHypRef Expression
1 funopab 6577 . . 3 (Fun {⟨𝑥, 𝑦⟩ ∣ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥 dom 𝑥))) ∨ (Lim dom 𝑥𝑦 = ran 𝑥))} ↔ ∀𝑥∃*𝑦((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥 dom 𝑥))) ∨ (Lim dom 𝑥𝑦 = ran 𝑥)))
2 fvex 6898 . . . 4 (𝐻‘(𝑥 dom 𝑥)) ∈ V
3 vex 3472 . . . . 5 𝑥 ∈ V
4 rnexg 7892 . . . . 5 (𝑥 ∈ V → ran 𝑥 ∈ V)
5 uniexg 7727 . . . . 5 (ran 𝑥 ∈ V → ran 𝑥 ∈ V)
63, 4, 5mp2b 10 . . . 4 ran 𝑥 ∈ V
7 nlim0 6417 . . . . . 6 ¬ Lim ∅
8 dm0 5914 . . . . . . 7 dom ∅ = ∅
9 limeq 6370 . . . . . . 7 (dom ∅ = ∅ → (Lim dom ∅ ↔ Lim ∅))
108, 9ax-mp 5 . . . . . 6 (Lim dom ∅ ↔ Lim ∅)
117, 10mtbir 323 . . . . 5 ¬ Lim dom ∅
12 dmeq 5897 . . . . . . 7 (𝑥 = ∅ → dom 𝑥 = dom ∅)
13 limeq 6370 . . . . . . 7 (dom 𝑥 = dom ∅ → (Lim dom 𝑥 ↔ Lim dom ∅))
1412, 13syl 17 . . . . . 6 (𝑥 = ∅ → (Lim dom 𝑥 ↔ Lim dom ∅))
1514biimpa 476 . . . . 5 ((𝑥 = ∅ ∧ Lim dom 𝑥) → Lim dom ∅)
1611, 15mto 196 . . . 4 ¬ (𝑥 = ∅ ∧ Lim dom 𝑥)
172, 6, 16moeq3 3703 . . 3 ∃*𝑦((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥 dom 𝑥))) ∨ (Lim dom 𝑥𝑦 = ran 𝑥))
181, 17mpgbir 1793 . 2 Fun {⟨𝑥, 𝑦⟩ ∣ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥 dom 𝑥))) ∨ (Lim dom 𝑥𝑦 = ran 𝑥))}
19 tz7.44lem1.1 . . 3 𝐺 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥 dom 𝑥))) ∨ (Lim dom 𝑥𝑦 = ran 𝑥))}
2019funeqi 6563 . 2 (Fun 𝐺 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥 dom 𝑥))) ∨ (Lim dom 𝑥𝑦 = ran 𝑥))})
2118, 20mpbir 230 1 Fun 𝐺
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 395  wo 844  w3o 1083   = wceq 1533  wcel 2098  ∃*wmo 2526  Vcvv 3468  c0 4317   cuni 4902  {copab 5203  dom cdm 5669  ran crn 5670  Lim wlim 6359  Fun wfun 6531  cfv 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-ord 6361  df-lim 6363  df-iota 6489  df-fun 6539  df-fv 6545
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator