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Theorem tz7.44lem1 8141
Description: 𝐺 is a function. Lemma for tz7.44-1 8142, tz7.44-2 8143, and tz7.44-3 8144. (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 6415 . . 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 6730 . . . 4 (𝐻‘(𝑥 dom 𝑥)) ∈ V
3 vex 3412 . . . . 5 𝑥 ∈ V
4 rnexg 7682 . . . . 5 (𝑥 ∈ V → ran 𝑥 ∈ V)
5 uniexg 7528 . . . . 5 (ran 𝑥 ∈ V → ran 𝑥 ∈ V)
63, 4, 5mp2b 10 . . . 4 ran 𝑥 ∈ V
7 nlim0 6271 . . . . . 6 ¬ Lim ∅
8 dm0 5789 . . . . . . 7 dom ∅ = ∅
9 limeq 6225 . . . . . . 7 (dom ∅ = ∅ → (Lim dom ∅ ↔ Lim ∅))
108, 9ax-mp 5 . . . . . 6 (Lim dom ∅ ↔ Lim ∅)
117, 10mtbir 326 . . . . 5 ¬ Lim dom ∅
12 dmeq 5772 . . . . . . 7 (𝑥 = ∅ → dom 𝑥 = dom ∅)
13 limeq 6225 . . . . . . 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 3625 . . 3 ∃*𝑦((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥 dom 𝑥))) ∨ (Lim dom 𝑥𝑦 = ran 𝑥))
181, 17mpgbir 1807 . 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 6401 . 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 847  w3o 1088   = wceq 1543  wcel 2110  ∃*wmo 2537  Vcvv 3408  c0 4237   cuni 4819  {copab 5115  dom cdm 5551  ran crn 5552  Lim wlim 6214  Fun wfun 6374  cfv 6380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-ord 6216  df-lim 6218  df-iota 6338  df-fun 6382  df-fv 6388
This theorem is referenced by: (None)
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