Theorem tz7.44lem1 8324

Description: The ordered pair abstraction 𝐺 defined in the hypothesis is a function. This was a lemma for tz7.44-1 8325, tz7.44-2 8326, and tz7.44-3 8327 when they used that definition of 𝐺. Now, they use the maps-to df-mpt 5171 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 6516	. . 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 6835	. . . 4 ⊢ (𝐻‘(𝑥‘∪ dom 𝑥)) ∈ V
3		vex 3440	. . . . 5 ⊢ 𝑥 ∈ V
4		rnexg 7832	. . . . 5 ⊢ (𝑥 ∈ V → ran 𝑥 ∈ V)
5		uniexg 7673	. . . . 5 ⊢ (ran 𝑥 ∈ V → ∪ ran 𝑥 ∈ V)
6	3, 4, 5	mp2b 10	. . . 4 ⊢ ∪ ran 𝑥 ∈ V
7		nlim0 6366	. . . . . 6 ⊢ ¬ Lim ∅
8		dm0 5859	. . . . . . 7 ⊢ dom ∅ = ∅
9		limeq 6318	. . . . . . 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 5842	. . . . . . 7 ⊢ (𝑥 = ∅ → dom 𝑥 = dom ∅)
13		limeq 6318	. . . . . . 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 3666	. . 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 6502	. 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 1541   ∈ wcel 2111  ∃*wmo 2533  Vcvv 3436  ∅c0 4280  ∪ cuni 4856  {copab 5151  dom cdm 5614  ran crn 5615  Lim wlim 6307  Fun wfun 6475  ‘cfv 6481

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-ord 6309  df-lim 6311  df-iota 6437  df-fun 6483  df-fv 6489

This theorem is referenced by: (None)