Theorem tz7.44lem1 8424

Description: The ordered pair abstraction 𝐺 defined in the hypothesis is a function. This was a lemma for tz7.44-1 8425, tz7.44-2 8426, and tz7.44-3 8427 when they used that definition of 𝐺. Now, they use the maps-to df-mpt 5207 idiom so this lemma is not needed anymore, but is kept in case other applications (for instance in intuitionistic set theory) need it. (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 6576	. . 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 6894	. . . 4 ⊢ (𝐻‘(𝑥‘∪ dom 𝑥)) ∈ V
3		vex 3468	. . . . 5 ⊢ 𝑥 ∈ V
4		rnexg 7903	. . . . 5 ⊢ (𝑥 ∈ V → ran 𝑥 ∈ V)
5		uniexg 7739	. . . . 5 ⊢ (ran 𝑥 ∈ V → ∪ ran 𝑥 ∈ V)
6	3, 4, 5	mp2b 10	. . . 4 ⊢ ∪ ran 𝑥 ∈ V
7		nlim0 6417	. . . . . 6 ⊢ ¬ Lim ∅
8		dm0 5905	. . . . . . 7 ⊢ dom ∅ = ∅
9		limeq 6369	. . . . . . 7 ⊢ (dom ∅ = ∅ → (Lim dom ∅ ↔ Lim ∅))
10	8, 9	ax-mp 5	. . . . . 6 ⊢ (Lim dom ∅ ↔ Lim ∅)
11	7, 10	mtbir 323	. . . . 5 ⊢ ¬ Lim dom ∅
12		dmeq 5888	. . . . . . 7 ⊢ (𝑥 = ∅ → dom 𝑥 = dom ∅)
13		limeq 6369	. . . . . . 7 ⊢ (dom 𝑥 = dom ∅ → (Lim dom 𝑥 ↔ Lim dom ∅))
14	12, 13	syl 17	. . . . . 6 ⊢ (𝑥 = ∅ → (Lim dom 𝑥 ↔ Lim dom ∅))
15	14	biimpa 476	. . . . 5 ⊢ ((𝑥 = ∅ ∧ Lim dom 𝑥) → Lim dom ∅)
16	11, 15	mto 197	. . . 4 ⊢ ¬ (𝑥 = ∅ ∧ Lim dom 𝑥)
17	2, 6, 16	moeq3 3700	. . 3 ⊢ ∃*𝑦((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥))
18	1, 17	mpgbir 1799	. 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 6562	. 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 231	1 ⊢ Fun 𝐺

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   = wceq 1540   ∈ wcel 2109  ∃*wmo 2538  Vcvv 3464  ∅c0 4313  ∪ cuni 4888  {copab 5186  dom cdm 5659  ran crn 5660  Lim wlim 6358  Fun wfun 6530  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-ord 6360  df-lim 6362  df-iota 6489  df-fun 6538  df-fv 6544

This theorem is referenced by: (None)