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Theorem tz7.44lem1 8026
 Description: 𝐺 is a function. Lemma for tz7.44-1 8027, tz7.44-2 8028, and tz7.44-3 8029. (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 6359 . . 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 6658 . . . 4 (𝐻‘(𝑥 dom 𝑥)) ∈ V
3 vex 3444 . . . . 5 𝑥 ∈ V
4 rnexg 7597 . . . . 5 (𝑥 ∈ V → ran 𝑥 ∈ V)
5 uniexg 7448 . . . . 5 (ran 𝑥 ∈ V → ran 𝑥 ∈ V)
63, 4, 5mp2b 10 . . . 4 ran 𝑥 ∈ V
7 nlim0 6217 . . . . . 6 ¬ Lim ∅
8 dm0 5754 . . . . . . 7 dom ∅ = ∅
9 limeq 6171 . . . . . . 7 (dom ∅ = ∅ → (Lim dom ∅ ↔ Lim ∅))
108, 9ax-mp 5 . . . . . 6 (Lim dom ∅ ↔ Lim ∅)
117, 10mtbir 326 . . . . 5 ¬ Lim dom ∅
12 dmeq 5736 . . . . . . 7 (𝑥 = ∅ → dom 𝑥 = dom ∅)
13 limeq 6171 . . . . . . 7 (dom 𝑥 = dom ∅ → (Lim dom 𝑥 ↔ Lim dom ∅))
1412, 13syl 17 . . . . . 6 (𝑥 = ∅ → (Lim dom 𝑥 ↔ Lim dom ∅))
1514biimpa 480 . . . . 5 ((𝑥 = ∅ ∧ Lim dom 𝑥) → Lim dom ∅)
1611, 15mto 200 . . . 4 ¬ (𝑥 = ∅ ∧ Lim dom 𝑥)
172, 6, 16moeq3 3651 . . 3 ∃*𝑦((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥 dom 𝑥))) ∨ (Lim dom 𝑥𝑦 = ran 𝑥))
181, 17mpgbir 1801 . 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 6345 . 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 234 1 Fun 𝐺
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∨ w3o 1083   = wceq 1538   ∈ wcel 2111  ∃*wmo 2596  Vcvv 3441  ∅c0 4243  ∪ cuni 4800  {copab 5092  dom cdm 5519  ran crn 5520  Lim wlim 6160  Fun wfun 6318  ‘cfv 6324 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-ord 6162  df-lim 6164  df-iota 6283  df-fun 6326  df-fv 6332 This theorem is referenced by: (None)
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