Theorem bnj1501 31515

Description: Technical lemma for bnj1500 31516. 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 490	. . . . . . . 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 31510	. . . . . . . . . 10 ⊢ (𝑅 FrSe 𝐴 → 𝐹 Fn 𝐴)
8		fndm 6168	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
9	7, 8	syl 17	. . . . . . . . 9 ⊢ (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
10	1, 9	bnj832 31208	. . . . . . . 8 ⊢ (𝜑 → dom 𝐹 = 𝐴)
11	2, 10	eleqtrrd 2847	. . . . . . 7 ⊢ (𝜑 → 𝑥 ∈ dom 𝐹)
12	6	dmeqi 5493	. . . . . . . 8 ⊢ dom 𝐹 = dom ∪ 𝐶
13	5	bnj1317 31272	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐶 → ∀𝑓 𝑤 ∈ 𝐶)
14	13	bnj1400 31286	. . . . . . . 8 ⊢ dom ∪ 𝐶 = ∪ 𝑓 ∈ 𝐶 dom 𝑓
15	12, 14	eqtri 2787	. . . . . . 7 ⊢ dom 𝐹 = ∪ 𝑓 ∈ 𝐶 dom 𝑓
16	11, 15	syl6eleq 2854	. . . . . 6 ⊢ (𝜑 → 𝑥 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓)
17	16	bnj1405 31287	. . . . 5 ⊢ (𝜑 → ∃𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓)
18		bnj1501.6	. . . . 5 ⊢ (𝜓 ↔ (𝜑 ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓))
19	17, 18	bnj1209 31247	. . . 4 ⊢ (𝜑 → ∃𝑓𝜓)
20	5	bnj1436 31290	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
21	20	bnj1299 31269	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 𝑓 Fn 𝑑)
22		fndm 6168	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
23	21, 22	bnj31 31168	. . . . . . . 8 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 dom 𝑓 = 𝑑)
24	18, 23	bnj836 31210	. . . . . . 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 31512	. . . . . . 7 ⊢ (𝜓 → ∀𝑑𝜓)
27	24, 25, 26	bnj1521 31301	. . . . . 6 ⊢ (𝜓 → ∃𝑑𝜒)
28	7	bnj930 31220	. . . . . . . . . . . 12 ⊢ (𝑅 FrSe 𝐴 → Fun 𝐹)
29	1, 28	bnj832 31208	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐹)
30	18, 29	bnj835 31209	. . . . . . . . . 10 ⊢ (𝜓 → Fun 𝐹)
31		elssuni 4625	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝐶 → 𝑓 ⊆ ∪ 𝐶)
32	31, 6	syl6sseqr 3812	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝐶 → 𝑓 ⊆ 𝐹)
33	18, 32	bnj836 31210	. . . . . . . . . 10 ⊢ (𝜓 → 𝑓 ⊆ 𝐹)
34	18	simp3bi 1177	. . . . . . . . . 10 ⊢ (𝜓 → 𝑥 ∈ dom 𝑓)
35	30, 33, 34	bnj1502 31298	. . . . . . . . 9 ⊢ (𝜓 → (𝐹‘𝑥) = (𝑓‘𝑥))
36	3, 4, 5	bnj1514 31511	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝐶 → ∀𝑥 ∈ dom 𝑓(𝑓‘𝑥) = (𝐺‘𝑌))
37	18, 36	bnj836 31210	. . . . . . . . . 10 ⊢ (𝜓 → ∀𝑥 ∈ dom 𝑓(𝑓‘𝑥) = (𝐺‘𝑌))
38	37, 34	bnj1294 31268	. . . . . . . . 9 ⊢ (𝜓 → (𝑓‘𝑥) = (𝐺‘𝑌))
39	35, 38	eqtrd 2799	. . . . . . . 8 ⊢ (𝜓 → (𝐹‘𝑥) = (𝐺‘𝑌))
40	25, 39	bnj835 31209	. . . . . . 7 ⊢ (𝜒 → (𝐹‘𝑥) = (𝐺‘𝑌))
41	25, 30	bnj835 31209	. . . . . . . . . . 11 ⊢ (𝜒 → Fun 𝐹)
42	25, 33	bnj835 31209	. . . . . . . . . . 11 ⊢ (𝜒 → 𝑓 ⊆ 𝐹)
43	3	bnj1517 31300	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ 𝐵 → ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
44	25, 43	bnj836 31210	. . . . . . . . . . . . 13 ⊢ (𝜒 → ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
45	25, 34	bnj835 31209	. . . . . . . . . . . . . 14 ⊢ (𝜒 → 𝑥 ∈ dom 𝑓)
46	25	simp3bi 1177	. . . . . . . . . . . . . 14 ⊢ (𝜒 → dom 𝑓 = 𝑑)
47	45, 46	eleqtrd 2846	. . . . . . . . . . . . 13 ⊢ (𝜒 → 𝑥 ∈ 𝑑)
48	44, 47	bnj1294 31268	. . . . . . . . . . . 12 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
49	48, 46	sseqtr4d 3802	. . . . . . . . . . 11 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑓)
50	41, 42, 49	bnj1503 31299	. . . . . . . . . 10 ⊢ (𝜒 → (𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)))
51	50	opeq2d 4566	. . . . . . . . 9 ⊢ (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
52	51, 4	syl6eqr 2817	. . . . . . . 8 ⊢ (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = 𝑌)
53	52	fveq2d 6379	. . . . . . 7 ⊢ (𝜒 → (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) = (𝐺‘𝑌))
54	40, 53	eqtr4d 2802	. . . . . 6 ⊢ (𝜒 → (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
55	27, 54	bnj593 31195	. . . . 5 ⊢ (𝜓 → ∃𝑑(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
56	3, 4, 5, 6	bnj1519 31513	. . . . 5 ⊢ ((𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑑(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
57	55, 56	bnj1397 31285	. . . 4 ⊢ (𝜓 → (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
58	19, 57	bnj593 31195	. . 3 ⊢ (𝜑 → ∃𝑓(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
59	3, 4, 5, 6	bnj1520 31514	. . 3 ⊢ ((𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑓(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
60	58, 59	bnj1397 31285	. 2 ⊢ (𝜑 → (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
61	1, 60	bnj1459 31293	1 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))

Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384   ∧ w3a 1107   = wceq 1652   ∈ wcel 2155  {cab 2751  ∀wral 3055  ∃wrex 3056   ⊆ wss 3732  ⟨cop 4340  ∪ cuni 4594  ∪ ciun 4676  dom cdm 5277   ↾ cres 5279  Fun wfun 6062   Fn wfn 6063  ‘cfv 6068   predc-bnj14 31137   FrSe w-bnj15 31141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-reg 8704  ax-inf2 8753

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-om 7264  df-1o 7764  df-bnj17 31136  df-bnj14 31138  df-bnj13 31140  df-bnj15 31142  df-bnj18 31144  df-bnj19 31146

This theorem is referenced by:  bnj1500  31516