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Theorem tz7.44lem1 7546
 Description: 𝐺 is a function. Lemma for tz7.44-1 7547, tz7.44-2 7548, and tz7.44-3 7549. (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 5961 . . 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 6239 . . . 4 (𝐻‘(𝑥 dom 𝑥)) ∈ V
3 vex 3234 . . . . 5 𝑥 ∈ V
4 rnexg 7140 . . . . 5 (𝑥 ∈ V → ran 𝑥 ∈ V)
5 uniexg 6997 . . . . 5 (ran 𝑥 ∈ V → ran 𝑥 ∈ V)
63, 4, 5mp2b 10 . . . 4 ran 𝑥 ∈ V
7 nlim0 5821 . . . . . 6 ¬ Lim ∅
8 dm0 5371 . . . . . . 7 dom ∅ = ∅
9 limeq 5773 . . . . . . 7 (dom ∅ = ∅ → (Lim dom ∅ ↔ Lim ∅))
108, 9ax-mp 5 . . . . . 6 (Lim dom ∅ ↔ Lim ∅)
117, 10mtbir 312 . . . . 5 ¬ Lim dom ∅
12 dmeq 5356 . . . . . . 7 (𝑥 = ∅ → dom 𝑥 = dom ∅)
13 limeq 5773 . . . . . . 7 (dom 𝑥 = dom ∅ → (Lim dom 𝑥 ↔ Lim dom ∅))
1412, 13syl 17 . . . . . 6 (𝑥 = ∅ → (Lim dom 𝑥 ↔ Lim dom ∅))
1514biimpa 500 . . . . 5 ((𝑥 = ∅ ∧ Lim dom 𝑥) → Lim dom ∅)
1611, 15mto 188 . . . 4 ¬ (𝑥 = ∅ ∧ Lim dom 𝑥)
172, 6, 16moeq3 3416 . . 3 ∃*𝑦((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥 dom 𝑥))) ∨ (Lim dom 𝑥𝑦 = ran 𝑥))
181, 17mpgbir 1766 . 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 5947 . 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 221 1 Fun 𝐺
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   ∨ wo 382   ∧ wa 383   ∨ w3o 1053   = wceq 1523   ∈ wcel 2030  ∃*wmo 2499  Vcvv 3231  ∅c0 3948  ∪ cuni 4468  {copab 4745  dom cdm 5143  ran crn 5144  Lim wlim 5762  Fun wfun 5920  ‘cfv 5926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-ord 5764  df-lim 5766  df-iota 5889  df-fun 5928  df-fv 5934 This theorem is referenced by: (None)
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