Theorem tz7.44lem1 8043

Description: 𝐺 is a function. Lemma for tz7.44-1 8044, tz7.44-2 8045, and tz7.44-3 8046. (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 𝑥 ∧ 𝑦 = ∪ 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

Step	Hyp	Ref	Expression
1		funopab 6392	. . 3 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥))} ↔ ∀𝑥∃*𝑦((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥)))
2		fvex 6685	. . . 4 ⊢ (𝐻‘(𝑥‘∪ dom 𝑥)) ∈ V
3		vex 3499	. . . . 5 ⊢ 𝑥 ∈ V
4		rnexg 7616	. . . . 5 ⊢ (𝑥 ∈ V → ran 𝑥 ∈ V)
5		uniexg 7468	. . . . 5 ⊢ (ran 𝑥 ∈ V → ∪ ran 𝑥 ∈ V)
6	3, 4, 5	mp2b 10	. . . 4 ⊢ ∪ ran 𝑥 ∈ V
7		nlim0 6251	. . . . . 6 ⊢ ¬ Lim ∅
8		dm0 5792	. . . . . . 7 ⊢ dom ∅ = ∅
9		limeq 6205	. . . . . . 7 ⊢ (dom ∅ = ∅ → (Lim dom ∅ ↔ Lim ∅))
10	8, 9	ax-mp 5	. . . . . 6 ⊢ (Lim dom ∅ ↔ Lim ∅)
11	7, 10	mtbir 325	. . . . 5 ⊢ ¬ Lim dom ∅
12		dmeq 5774	. . . . . . 7 ⊢ (𝑥 = ∅ → dom 𝑥 = dom ∅)
13		limeq 6205	. . . . . . 7 ⊢ (dom 𝑥 = dom ∅ → (Lim dom 𝑥 ↔ Lim dom ∅))
14	12, 13	syl 17	. . . . . 6 ⊢ (𝑥 = ∅ → (Lim dom 𝑥 ↔ Lim dom ∅))
15	14	biimpa 479	. . . . 5 ⊢ ((𝑥 = ∅ ∧ Lim dom 𝑥) → Lim dom ∅)
16	11, 15	mto 199	. . . 4 ⊢ ¬ (𝑥 = ∅ ∧ Lim dom 𝑥)
17	2, 6, 16	moeq3 3705	. . 3 ⊢ ∃*𝑦((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥))
18	1, 17	mpgbir 1800	. 2 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥))}
19		tz7.44lem1.1	. . 3 ⊢ 𝐺 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥))}
20	19	funeqi 6378	. 2 ⊢ (Fun 𝐺 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥))})
21	18, 20	mpbir 233	1 ⊢ Fun 𝐺

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∨ w3o 1082   = wceq 1537   ∈ wcel 2114  ∃*wmo 2620  Vcvv 3496  ∅c0 4293  ∪ cuni 4840  {copab 5130  dom cdm 5557  ran crn 5558  Lim wlim 6194  Fun wfun 6351  ‘cfv 6357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-ord 6196  df-lim 6198  df-iota 6316  df-fun 6359  df-fv 6365

This theorem is referenced by: (None)