Theorem tz7.44lem1 8422

Description: The ordered pair abstraction 𝐺 defined in the hypothesis is a function. This was a lemma for tz7.44-1 8423, tz7.44-2 8424, and tz7.44-3 8425 when they used that definition of 𝐺. Now, they use the maps-to df-mpt 5227 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 6582	. . 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 6904	. . . 4 ⊢ (𝐻‘(𝑥‘∪ dom 𝑥)) ∈ V
3		vex 3467	. . . . 5 ⊢ 𝑥 ∈ V
4		rnexg 7906	. . . . 5 ⊢ (𝑥 ∈ V → ran 𝑥 ∈ V)
5		uniexg 7742	. . . . 5 ⊢ (ran 𝑥 ∈ V → ∪ ran 𝑥 ∈ V)
6	3, 4, 5	mp2b 10	. . . 4 ⊢ ∪ ran 𝑥 ∈ V
7		nlim0 6423	. . . . . 6 ⊢ ¬ Lim ∅
8		dm0 5917	. . . . . . 7 ⊢ dom ∅ = ∅
9		limeq 6376	. . . . . . 7 ⊢ (dom ∅ = ∅ → (Lim dom ∅ ↔ Lim ∅))
10	8, 9	ax-mp 5	. . . . . 6 ⊢ (Lim dom ∅ ↔ Lim ∅)
11	7, 10	mtbir 322	. . . . 5 ⊢ ¬ Lim dom ∅
12		dmeq 5900	. . . . . . 7 ⊢ (𝑥 = ∅ → dom 𝑥 = dom ∅)
13		limeq 6376	. . . . . . 7 ⊢ (dom 𝑥 = dom ∅ → (Lim dom 𝑥 ↔ Lim dom ∅))
14	12, 13	syl 17	. . . . . 6 ⊢ (𝑥 = ∅ → (Lim dom 𝑥 ↔ Lim dom ∅))
15	14	biimpa 475	. . . . 5 ⊢ ((𝑥 = ∅ ∧ Lim dom 𝑥) → Lim dom ∅)
16	11, 15	mto 196	. . . 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 1793	. 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 6568	. 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 230	1 ⊢ Fun 𝐺

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 394   ∨ wo 845   ∨ w3o 1083   = wceq 1533   ∈ wcel 2098  ∃*wmo 2526  Vcvv 3463  ∅c0 4318  ∪ cuni 4903  {copab 5205  dom cdm 5672  ran crn 5673  Lim wlim 6365  Fun wfun 6536  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7737

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-ord 6367  df-lim 6369  df-iota 6494  df-fun 6544  df-fv 6550

This theorem is referenced by: (None)