Theorem tz7.44lem1 8348

Description: The ordered pair abstraction 𝐺 defined in the hypothesis is a function. This was a lemma for tz7.44-1 8349, tz7.44-2 8350, and tz7.44-3 8351 when they used that definition of 𝐺. Now, they use the maps-to df-mpt 5182 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 6537	. . 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 6857	. . . 4 ⊢ (𝐻‘(𝑥‘∪ dom 𝑥)) ∈ V
3		vex 3446	. . . . 5 ⊢ 𝑥 ∈ V
4		rnexg 7856	. . . . 5 ⊢ (𝑥 ∈ V → ran 𝑥 ∈ V)
5		uniexg 7697	. . . . 5 ⊢ (ran 𝑥 ∈ V → ∪ ran 𝑥 ∈ V)
6	3, 4, 5	mp2b 10	. . . 4 ⊢ ∪ ran 𝑥 ∈ V
7		nlim0 6387	. . . . . 6 ⊢ ¬ Lim ∅
8		dm0 5879	. . . . . . 7 ⊢ dom ∅ = ∅
9		limeq 6339	. . . . . . 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 5862	. . . . . . 7 ⊢ (𝑥 = ∅ → dom 𝑥 = dom ∅)
13		limeq 6339	. . . . . . 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 3672	. . 3 ⊢ ∃*𝑦((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥))
18	1, 17	mpgbir 1801	. 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 6523	. 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 848   ∨ w3o 1086   = wceq 1542   ∈ wcel 2114  ∃*wmo 2538  Vcvv 3442  ∅c0 4287  ∪ cuni 4865  {copab 5162  dom cdm 5634  ran crn 5635  Lim wlim 6328  Fun wfun 6496  ‘cfv 6502

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-ord 6330  df-lim 6332  df-iota 6458  df-fun 6504  df-fv 6510

This theorem is referenced by: (None)