Theorem bnj1501 31135

Description: Technical lemma for bnj1500 31136. 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 480	. . . . . . . 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 31130	. . . . . . . . . 10 ⊢ (𝑅 FrSe 𝐴 → 𝐹 Fn 𝐴)
8		fndm 5990	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
9	7, 8	syl 17	. . . . . . . . 9 ⊢ (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
10	1, 9	bnj832 30828	. . . . . . . 8 ⊢ (𝜑 → dom 𝐹 = 𝐴)
11	2, 10	eleqtrrd 2704	. . . . . . 7 ⊢ (𝜑 → 𝑥 ∈ dom 𝐹)
12	6	dmeqi 5325	. . . . . . . 8 ⊢ dom 𝐹 = dom ∪ 𝐶
13	5	bnj1317 30892	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐶 → ∀𝑓 𝑤 ∈ 𝐶)
14	13	bnj1400 30906	. . . . . . . 8 ⊢ dom ∪ 𝐶 = ∪ 𝑓 ∈ 𝐶 dom 𝑓
15	12, 14	eqtri 2644	. . . . . . 7 ⊢ dom 𝐹 = ∪ 𝑓 ∈ 𝐶 dom 𝑓
16	11, 15	syl6eleq 2711	. . . . . 6 ⊢ (𝜑 → 𝑥 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓)
17	16	bnj1405 30907	. . . . 5 ⊢ (𝜑 → ∃𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓)
18		bnj1501.6	. . . . 5 ⊢ (𝜓 ↔ (𝜑 ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓))
19	17, 18	bnj1209 30867	. . . 4 ⊢ (𝜑 → ∃𝑓𝜓)
20	5	bnj1436 30910	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
21	20	bnj1299 30889	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 𝑓 Fn 𝑑)
22		fndm 5990	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
23	21, 22	bnj31 30785	. . . . . . . 8 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 dom 𝑓 = 𝑑)
24	18, 23	bnj836 30830	. . . . . . 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 31132	. . . . . . 7 ⊢ (𝜓 → ∀𝑑𝜓)
27	24, 25, 26	bnj1521 30921	. . . . . 6 ⊢ (𝜓 → ∃𝑑𝜒)
28	7	bnj930 30840	. . . . . . . . . . . 12 ⊢ (𝑅 FrSe 𝐴 → Fun 𝐹)
29	1, 28	bnj832 30828	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐹)
30	18, 29	bnj835 30829	. . . . . . . . . 10 ⊢ (𝜓 → Fun 𝐹)
31		elssuni 4467	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝐶 → 𝑓 ⊆ ∪ 𝐶)
32	31, 6	syl6sseqr 3652	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝐶 → 𝑓 ⊆ 𝐹)
33	18, 32	bnj836 30830	. . . . . . . . . 10 ⊢ (𝜓 → 𝑓 ⊆ 𝐹)
34	18	simp3bi 1078	. . . . . . . . . 10 ⊢ (𝜓 → 𝑥 ∈ dom 𝑓)
35	30, 33, 34	bnj1502 30918	. . . . . . . . 9 ⊢ (𝜓 → (𝐹‘𝑥) = (𝑓‘𝑥))
36	3, 4, 5	bnj1514 31131	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝐶 → ∀𝑥 ∈ dom 𝑓(𝑓‘𝑥) = (𝐺‘𝑌))
37	18, 36	bnj836 30830	. . . . . . . . . 10 ⊢ (𝜓 → ∀𝑥 ∈ dom 𝑓(𝑓‘𝑥) = (𝐺‘𝑌))
38	37, 34	bnj1294 30888	. . . . . . . . 9 ⊢ (𝜓 → (𝑓‘𝑥) = (𝐺‘𝑌))
39	35, 38	eqtrd 2656	. . . . . . . 8 ⊢ (𝜓 → (𝐹‘𝑥) = (𝐺‘𝑌))
40	25, 39	bnj835 30829	. . . . . . 7 ⊢ (𝜒 → (𝐹‘𝑥) = (𝐺‘𝑌))
41	25, 30	bnj835 30829	. . . . . . . . . . 11 ⊢ (𝜒 → Fun 𝐹)
42	25, 33	bnj835 30829	. . . . . . . . . . 11 ⊢ (𝜒 → 𝑓 ⊆ 𝐹)
43	3	bnj1517 30920	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ 𝐵 → ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
44	25, 43	bnj836 30830	. . . . . . . . . . . . 13 ⊢ (𝜒 → ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
45	25, 34	bnj835 30829	. . . . . . . . . . . . . 14 ⊢ (𝜒 → 𝑥 ∈ dom 𝑓)
46	25	simp3bi 1078	. . . . . . . . . . . . . 14 ⊢ (𝜒 → dom 𝑓 = 𝑑)
47	45, 46	eleqtrd 2703	. . . . . . . . . . . . 13 ⊢ (𝜒 → 𝑥 ∈ 𝑑)
48	44, 47	bnj1294 30888	. . . . . . . . . . . 12 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
49	48, 46	sseqtr4d 3642	. . . . . . . . . . 11 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑓)
50	41, 42, 49	bnj1503 30919	. . . . . . . . . 10 ⊢ (𝜒 → (𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)))
51	50	opeq2d 4409	. . . . . . . . 9 ⊢ (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
52	51, 4	syl6eqr 2674	. . . . . . . 8 ⊢ (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = 𝑌)
53	52	fveq2d 6195	. . . . . . 7 ⊢ (𝜒 → (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) = (𝐺‘𝑌))
54	40, 53	eqtr4d 2659	. . . . . 6 ⊢ (𝜒 → (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
55	27, 54	bnj593 30815	. . . . 5 ⊢ (𝜓 → ∃𝑑(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
56	3, 4, 5, 6	bnj1519 31133	. . . . 5 ⊢ ((𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑑(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
57	55, 56	bnj1397 30905	. . . 4 ⊢ (𝜓 → (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
58	19, 57	bnj593 30815	. . 3 ⊢ (𝜑 → ∃𝑓(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
59	3, 4, 5, 6	bnj1520 31134	. . 3 ⊢ ((𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑓(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
60	58, 59	bnj1397 30905	. 2 ⊢ (𝜑 → (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
61	1, 60	bnj1459 30913	1 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913   ⊆ wss 3574  ⟨cop 4183  ∪ cuni 4436  ∪ ciun 4520  dom cdm 5114   ↾ cres 5116  Fun wfun 5882   Fn wfn 5883  ‘cfv 5888   predc-bnj14 30754   FrSe w-bnj15 30758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-bnj17 30753  df-bnj14 30755  df-bnj13 30757  df-bnj15 30759  df-bnj18 30761  df-bnj19 30763

This theorem is referenced by:  bnj1500  31136