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Theorem tz7.44lem1 7900
Description: 𝐺 is a function. Lemma for tz7.44-1 7901, tz7.44-2 7902, and tz7.44-3 7903. (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 6267 . . 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 6558 . . . 4 (𝐻‘(𝑥 dom 𝑥)) ∈ V
3 vex 3443 . . . . 5 𝑥 ∈ V
4 rnexg 7477 . . . . 5 (𝑥 ∈ V → ran 𝑥 ∈ V)
5 uniexg 7332 . . . . 5 (ran 𝑥 ∈ V → ran 𝑥 ∈ V)
63, 4, 5mp2b 10 . . . 4 ran 𝑥 ∈ V
7 nlim0 6131 . . . . . 6 ¬ Lim ∅
8 dm0 5683 . . . . . . 7 dom ∅ = ∅
9 limeq 6085 . . . . . . 7 (dom ∅ = ∅ → (Lim dom ∅ ↔ Lim ∅))
108, 9ax-mp 5 . . . . . 6 (Lim dom ∅ ↔ Lim ∅)
117, 10mtbir 324 . . . . 5 ¬ Lim dom ∅
12 dmeq 5665 . . . . . . 7 (𝑥 = ∅ → dom 𝑥 = dom ∅)
13 limeq 6085 . . . . . . 7 (dom 𝑥 = dom ∅ → (Lim dom 𝑥 ↔ Lim dom ∅))
1412, 13syl 17 . . . . . 6 (𝑥 = ∅ → (Lim dom 𝑥 ↔ Lim dom ∅))
1514biimpa 477 . . . . 5 ((𝑥 = ∅ ∧ Lim dom 𝑥) → Lim dom ∅)
1611, 15mto 198 . . . 4 ¬ (𝑥 = ∅ ∧ Lim dom 𝑥)
172, 6, 16moeq3 3644 . . 3 ∃*𝑦((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥 dom 𝑥))) ∨ (Lim dom 𝑥𝑦 = ran 𝑥))
181, 17mpgbir 1785 . 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 6253 . 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 232 1 Fun 𝐺
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wa 396  wo 842  w3o 1079   = wceq 1525  wcel 2083  ∃*wmo 2576  Vcvv 3440  c0 4217   cuni 4751  {copab 5030  dom cdm 5450  ran crn 5451  Lim wlim 6074  Fun wfun 6226  cfv 6232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-ord 6076  df-lim 6078  df-iota 6196  df-fun 6234  df-fv 6240
This theorem is referenced by: (None)
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