Theorem bnj1501 32339

Description: Technical lemma for bnj1500 32340. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1501.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1501.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1501.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1501.4	⊢ 𝐹 = ∪ 𝐶
bnj1501.5	⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴))
bnj1501.6	⊢ (𝜓 ↔ (𝜑 ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓))
bnj1501.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑑 ∈ 𝐵 ∧ dom 𝑓 = 𝑑))

Assertion

Ref	Expression
bnj1501	⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))

Distinct variable groups:   𝐴,𝑑,𝑓,𝑥   𝐵,𝑓   𝐺,𝑑,𝑓,𝑥   𝑅,𝑑,𝑓,𝑥   𝑌,𝑑   𝜑,𝑑,𝑓

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑓,𝑑)   𝜒(𝑥,𝑓,𝑑)   𝐵(𝑥,𝑑)   𝐶(𝑥,𝑓,𝑑)   𝐹(𝑥,𝑓,𝑑)   𝑌(𝑥,𝑓)

Proof of Theorem bnj1501

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1501.5	. 2 ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴))
2	1	simprbi 499	. . . . . . . 8 ⊢ (𝜑 → 𝑥 ∈ 𝐴)
3		bnj1501.1	. . . . . . . . . . 11 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
4		bnj1501.2	. . . . . . . . . . 11 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
5		bnj1501.3	. . . . . . . . . . 11 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
6		bnj1501.4	. . . . . . . . . . 11 ⊢ 𝐹 = ∪ 𝐶
7	3, 4, 5, 6	bnj60 32334	. . . . . . . . . 10 ⊢ (𝑅 FrSe 𝐴 → 𝐹 Fn 𝐴)
8		fndm 6455	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
9	7, 8	syl 17	. . . . . . . . 9 ⊢ (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
10	1, 9	bnj832 32029	. . . . . . . 8 ⊢ (𝜑 → dom 𝐹 = 𝐴)
11	2, 10	eleqtrrd 2916	. . . . . . 7 ⊢ (𝜑 → 𝑥 ∈ dom 𝐹)
12	6	dmeqi 5773	. . . . . . . 8 ⊢ dom 𝐹 = dom ∪ 𝐶
13	5	bnj1317 32093	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐶 → ∀𝑓 𝑤 ∈ 𝐶)
14	13	bnj1400 32107	. . . . . . . 8 ⊢ dom ∪ 𝐶 = ∪ 𝑓 ∈ 𝐶 dom 𝑓
15	12, 14	eqtri 2844	. . . . . . 7 ⊢ dom 𝐹 = ∪ 𝑓 ∈ 𝐶 dom 𝑓
16	11, 15	eleqtrdi 2923	. . . . . 6 ⊢ (𝜑 → 𝑥 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓)
17	16	bnj1405 32108	. . . . 5 ⊢ (𝜑 → ∃𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓)
18		bnj1501.6	. . . . 5 ⊢ (𝜓 ↔ (𝜑 ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓))
19	17, 18	bnj1209 32068	. . . 4 ⊢ (𝜑 → ∃𝑓𝜓)
20	5	bnj1436 32111	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
21	20	bnj1299 32090	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 𝑓 Fn 𝑑)
22		fndm 6455	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
23	21, 22	bnj31 31989	. . . . . . . 8 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 dom 𝑓 = 𝑑)
24	18, 23	bnj836 32031	. . . . . . 7 ⊢ (𝜓 → ∃𝑑 ∈ 𝐵 dom 𝑓 = 𝑑)
25		bnj1501.7	. . . . . . 7 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑑 ∈ 𝐵 ∧ dom 𝑓 = 𝑑))
26	3, 4, 5, 6, 1, 18	bnj1518 32336	. . . . . . 7 ⊢ (𝜓 → ∀𝑑𝜓)
27	24, 25, 26	bnj1521 32123	. . . . . 6 ⊢ (𝜓 → ∃𝑑𝜒)
28	7	bnj930 32041	. . . . . . . . . . . 12 ⊢ (𝑅 FrSe 𝐴 → Fun 𝐹)
29	1, 28	bnj832 32029	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐹)
30	18, 29	bnj835 32030	. . . . . . . . . 10 ⊢ (𝜓 → Fun 𝐹)
31		elssuni 4868	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝐶 → 𝑓 ⊆ ∪ 𝐶)
32	31, 6	sseqtrrdi 4018	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝐶 → 𝑓 ⊆ 𝐹)
33	18, 32	bnj836 32031	. . . . . . . . . 10 ⊢ (𝜓 → 𝑓 ⊆ 𝐹)
34	18	simp3bi 1143	. . . . . . . . . 10 ⊢ (𝜓 → 𝑥 ∈ dom 𝑓)
35	30, 33, 34	bnj1502 32120	. . . . . . . . 9 ⊢ (𝜓 → (𝐹‘𝑥) = (𝑓‘𝑥))
36	3, 4, 5	bnj1514 32335	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝐶 → ∀𝑥 ∈ dom 𝑓(𝑓‘𝑥) = (𝐺‘𝑌))
37	18, 36	bnj836 32031	. . . . . . . . . 10 ⊢ (𝜓 → ∀𝑥 ∈ dom 𝑓(𝑓‘𝑥) = (𝐺‘𝑌))
38	37, 34	bnj1294 32089	. . . . . . . . 9 ⊢ (𝜓 → (𝑓‘𝑥) = (𝐺‘𝑌))
39	35, 38	eqtrd 2856	. . . . . . . 8 ⊢ (𝜓 → (𝐹‘𝑥) = (𝐺‘𝑌))
40	25, 39	bnj835 32030	. . . . . . 7 ⊢ (𝜒 → (𝐹‘𝑥) = (𝐺‘𝑌))
41	25, 30	bnj835 32030	. . . . . . . . . . 11 ⊢ (𝜒 → Fun 𝐹)
42	25, 33	bnj835 32030	. . . . . . . . . . 11 ⊢ (𝜒 → 𝑓 ⊆ 𝐹)
43	3	bnj1517 32122	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ 𝐵 → ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
44	25, 43	bnj836 32031	. . . . . . . . . . . . 13 ⊢ (𝜒 → ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
45	25, 34	bnj835 32030	. . . . . . . . . . . . . 14 ⊢ (𝜒 → 𝑥 ∈ dom 𝑓)
46	25	simp3bi 1143	. . . . . . . . . . . . . 14 ⊢ (𝜒 → dom 𝑓 = 𝑑)
47	45, 46	eleqtrd 2915	. . . . . . . . . . . . 13 ⊢ (𝜒 → 𝑥 ∈ 𝑑)
48	44, 47	bnj1294 32089	. . . . . . . . . . . 12 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
49	48, 46	sseqtrrd 4008	. . . . . . . . . . 11 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑓)
50	41, 42, 49	bnj1503 32121	. . . . . . . . . 10 ⊢ (𝜒 → (𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)))
51	50	opeq2d 4810	. . . . . . . . 9 ⊢ (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
52	51, 4	syl6eqr 2874	. . . . . . . 8 ⊢ (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = 𝑌)
53	52	fveq2d 6674	. . . . . . 7 ⊢ (𝜒 → (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) = (𝐺‘𝑌))
54	40, 53	eqtr4d 2859	. . . . . 6 ⊢ (𝜒 → (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
55	27, 54	bnj593 32016	. . . . 5 ⊢ (𝜓 → ∃𝑑(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
56	3, 4, 5, 6	bnj1519 32337	. . . . 5 ⊢ ((𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑑(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
57	55, 56	bnj1397 32106	. . . 4 ⊢ (𝜓 → (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
58	19, 57	bnj593 32016	. . 3 ⊢ (𝜑 → ∃𝑓(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
59	3, 4, 5, 6	bnj1520 32338	. . 3 ⊢ ((𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑓(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
60	58, 59	bnj1397 32106	. 2 ⊢ (𝜑 → (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
61	1, 60	bnj1459 32115	1 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  {cab 2799  ∀wral 3138  ∃wrex 3139   ⊆ wss 3936  ⟨cop 4573  ∪ cuni 4838  ∪ ciun 4919  dom cdm 5555   ↾ cres 5557  Fun wfun 6349   Fn wfn 6350  ‘cfv 6355   predc-bnj14 31958   FrSe w-bnj15 31962

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-reg 9056  ax-inf2 9104

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-om 7581  df-1o 8102  df-bnj17 31957  df-bnj14 31959  df-bnj13 31961  df-bnj15 31963  df-bnj18 31965  df-bnj19 31967

This theorem is referenced by:  bnj1500  32340