Theorem tz7.44-1 2966

Description: The value of F at ∅. Part 1 of Theorem 7.44 of [TakeutiZaring] p. 49.

Hypotheses

Ref	Expression
tz7.44.1	⊢ G = {⟨x, y⟩∣((x = ∅ ∧ y = A) ∨ (¬ (x = ∅ ∨ Lim dom x) ∧ y = (H ‘(x ‘∪dom x))) ∨ (Lim dom x ∧ y = ∪ran x))}
tz7.44.2	⊢ F Fn On
tz7.44.3	⊢ (x ∈ On → (F ‘x) = (G ‘(F ↾ x)))
tz7.44.4	⊢ A ∈ V

Assertion

Ref	Expression
tz7.44-1	⊢ (F ‘∅) = A

Distinct variable group(s):   x,y,A   x,F   x,G   y,H

Proof of Theorem tz7.44-1

Step	Hyp	Ref	Expression
1		0elon 2277	. . 3 ⊢ ∅ ∈ On
2		fveq2 2832	. . . . 5 ⊢ (x = ∅ → (F ‘x) = (F ‘∅))
3		reseq2 2576	. . . . . 6 ⊢ (x = ∅ → (F ↾ x) = (F ↾ ∅))
4	3	fveq2d 2836	. . . . 5 ⊢ (x = ∅ → (G ‘(F ↾ x)) = (G ‘(F ↾ ∅)))
5	2, 4	cleq12d 1115	. . . 4 ⊢ (x = ∅ → ((F ‘x) = (G ‘(F ↾ x)) ↔ (F ‘∅) = (G ‘(F ↾ ∅))))
6		tz7.44.3	. . . 4 ⊢ (x ∈ On → (F ‘x) = (G ‘(F ↾ x)))
7	5, 6	vtoclga 1387	. . 3 ⊢ (∅ ∈ On → (F ‘∅) = (G ‘(F ↾ ∅)))
8	1, 7	ax-mp 6	. 2 ⊢ (F ‘∅) = (G ‘(F ↾ ∅))
9		res0 2578	. . 3 ⊢ (F ↾ ∅) = ∅
10	9	fveq2i 2835	. 2 ⊢ (G ‘(F ↾ ∅)) = (G ‘∅)
11		tz7.44.1	. . . 4 ⊢ G = {⟨x, y⟩∣((x = ∅ ∧ y = A) ∨ (¬ (x = ∅ ∨ Lim dom x) ∧ y = (H ‘(x ‘∪dom x))) ∨ (Lim dom x ∧ y = ∪ran x))}
12	11	tz7.44lem1 2965	. . 3 ⊢ Fun G
13		3mix1 600	. . . . . 6 ⊢ ((x = ∅ ∧ y = A) → ((x = ∅ ∧ y = A) ∨ (¬ (x = ∅ ∨ Lim dom x) ∧ y = (H ‘(x ‘∪dom x))) ∨ (Lim dom x ∧ y = ∪ran x)))
14	13	ssopab2i 2120	. . . . 5 ⊢ {⟨x, y⟩∣(x = ∅ ∧ y = A)} ⊆ {⟨x, y⟩∣((x = ∅ ∧ y = A) ∨ (¬ (x = ∅ ∨ Lim dom x) ∧ y = (H ‘(x ‘∪dom x))) ∨ (Lim dom x ∧ y = ∪ran x))}
15	14, 11	sseqtr4 1533	. . . 4 ⊢ {⟨x, y⟩∣(x = ∅ ∧ y = A)} ⊆ G
16		cleqid 1102	. . . . . 6 ⊢ ∅ = ∅
17		cleqid 1102	. . . . . 6 ⊢ A = A
18	16, 17	pm3.2i 234	. . . . 5 ⊢ (∅ = ∅ ∧ A = A)
19		0ex 1745	. . . . . 6 ⊢ ∅ ∈ V
20		tz7.44.4	. . . . . 6 ⊢ A ∈ V
21		cleq1 1107	. . . . . . 7 ⊢ (x = ∅ → (x = ∅ ↔ ∅ = ∅))
22	21	anbi1d 469	. . . . . 6 ⊢ (x = ∅ → ((x = ∅ ∧ y = A) ↔ (∅ = ∅ ∧ y = A)))
23		cleq1 1107	. . . . . . 7 ⊢ (y = A → (y = A ↔ A = A))
24	23	anbi2d 468	. . . . . 6 ⊢ (y = A → ((∅ = ∅ ∧ y = A) ↔ (∅ = ∅ ∧ A = A)))
25	19, 20, 22, 24	opelopab 2117	. . . . 5 ⊢ (⟨∅, A⟩ ∈ {⟨x, y⟩∣(x = ∅ ∧ y = A)} ↔ (∅ = ∅ ∧ A = A))
26	18, 25	mpbir 165	. . . 4 ⊢ ⟨∅, A⟩ ∈ {⟨x, y⟩∣(x = ∅ ∧ y = A)}
27	15, 26	sselii 1505	. . 3 ⊢ ⟨∅, A⟩ ∈ G
28	20	funfvopi 2853	. . 3 ⊢ (Fun G → (⟨∅, A⟩ ∈ G → (G ‘∅) = A))
29	12, 27, 28	mp2 43	. 2 ⊢ (G ‘∅) = A
30	8, 10, 29	3eqtr 1123	1 ⊢ (F ‘∅) = A

Syntax hints:  ¬ wn 1   → wi 2   ∨ wo 195   ∧ wa 196   ∨ w3o 580   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∅c0 1707  ⟨cop 1810  ∪cuni 1919  {copab 2055  Oncon0 2199  Lim wlim 2200  dom cdm 2410  ran crn 2411   ↾ cres 2412  Fun wfun 2416   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  rdgzer 2979

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438

metamath.org