Theorem bnj1501 33026

Description: Technical lemma for bnj1500 33027. 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 496	. . . . . . . 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 33021	. . . . . . . . . 10 ⊢ (𝑅 FrSe 𝐴 → 𝐹 Fn 𝐴)
8	7	fndmd 6534	. . . . . . . . 9 ⊢ (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)
9	1, 8	bnj832 32717	. . . . . . . 8 ⊢ (𝜑 → dom 𝐹 = 𝐴)
10	2, 9	eleqtrrd 2843	. . . . . . 7 ⊢ (𝜑 → 𝑥 ∈ dom 𝐹)
11	6	dmeqi 5810	. . . . . . . 8 ⊢ dom 𝐹 = dom ∪ 𝐶
12	5	bnj1317 32780	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐶 → ∀𝑓 𝑤 ∈ 𝐶)
13	12	bnj1400 32794	. . . . . . . 8 ⊢ dom ∪ 𝐶 = ∪ 𝑓 ∈ 𝐶 dom 𝑓
14	11, 13	eqtri 2767	. . . . . . 7 ⊢ dom 𝐹 = ∪ 𝑓 ∈ 𝐶 dom 𝑓
15	10, 14	eleqtrdi 2850	. . . . . 6 ⊢ (𝜑 → 𝑥 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓)
16	15	bnj1405 32795	. . . . 5 ⊢ (𝜑 → ∃𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓)
17		bnj1501.6	. . . . 5 ⊢ (𝜓 ↔ (𝜑 ∧ 𝑓 ∈ 𝐶 ∧ 𝑥 ∈ dom 𝑓))
18	16, 17	bnj1209 32755	. . . 4 ⊢ (𝜑 → ∃𝑓𝜓)
19	5	bnj1436 32798	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
20	19	bnj1299 32777	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 𝑓 Fn 𝑑)
21		fndm 6532	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
22	20, 21	bnj31 32677	. . . . . . . 8 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 dom 𝑓 = 𝑑)
23	17, 22	bnj836 32719	. . . . . . 7 ⊢ (𝜓 → ∃𝑑 ∈ 𝐵 dom 𝑓 = 𝑑)
24		bnj1501.7	. . . . . . 7 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑑 ∈ 𝐵 ∧ dom 𝑓 = 𝑑))
25	3, 4, 5, 6, 1, 17	bnj1518 33023	. . . . . . 7 ⊢ (𝜓 → ∀𝑑𝜓)
26	23, 24, 25	bnj1521 32810	. . . . . 6 ⊢ (𝜓 → ∃𝑑𝜒)
27	7	fnfund 6530	. . . . . . . . . . . 12 ⊢ (𝑅 FrSe 𝐴 → Fun 𝐹)
28	1, 27	bnj832 32717	. . . . . . . . . . 11 ⊢ (𝜑 → Fun 𝐹)
29	17, 28	bnj835 32718	. . . . . . . . . 10 ⊢ (𝜓 → Fun 𝐹)
30		elssuni 4876	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝐶 → 𝑓 ⊆ ∪ 𝐶)
31	30, 6	sseqtrrdi 3976	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝐶 → 𝑓 ⊆ 𝐹)
32	17, 31	bnj836 32719	. . . . . . . . . 10 ⊢ (𝜓 → 𝑓 ⊆ 𝐹)
33	17	simp3bi 1145	. . . . . . . . . 10 ⊢ (𝜓 → 𝑥 ∈ dom 𝑓)
34	29, 32, 33	bnj1502 32807	. . . . . . . . 9 ⊢ (𝜓 → (𝐹‘𝑥) = (𝑓‘𝑥))
35	3, 4, 5	bnj1514 33022	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝐶 → ∀𝑥 ∈ dom 𝑓(𝑓‘𝑥) = (𝐺‘𝑌))
36	17, 35	bnj836 32719	. . . . . . . . . 10 ⊢ (𝜓 → ∀𝑥 ∈ dom 𝑓(𝑓‘𝑥) = (𝐺‘𝑌))
37	36, 33	bnj1294 32776	. . . . . . . . 9 ⊢ (𝜓 → (𝑓‘𝑥) = (𝐺‘𝑌))
38	34, 37	eqtrd 2779	. . . . . . . 8 ⊢ (𝜓 → (𝐹‘𝑥) = (𝐺‘𝑌))
39	24, 38	bnj835 32718	. . . . . . 7 ⊢ (𝜒 → (𝐹‘𝑥) = (𝐺‘𝑌))
40	24, 29	bnj835 32718	. . . . . . . . . . 11 ⊢ (𝜒 → Fun 𝐹)
41	24, 32	bnj835 32718	. . . . . . . . . . 11 ⊢ (𝜒 → 𝑓 ⊆ 𝐹)
42	3	bnj1517 32809	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ 𝐵 → ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
43	24, 42	bnj836 32719	. . . . . . . . . . . . 13 ⊢ (𝜒 → ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
44	24, 33	bnj835 32718	. . . . . . . . . . . . . 14 ⊢ (𝜒 → 𝑥 ∈ dom 𝑓)
45	24	simp3bi 1145	. . . . . . . . . . . . . 14 ⊢ (𝜒 → dom 𝑓 = 𝑑)
46	44, 45	eleqtrd 2842	. . . . . . . . . . . . 13 ⊢ (𝜒 → 𝑥 ∈ 𝑑)
47	43, 46	bnj1294 32776	. . . . . . . . . . . 12 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)
48	47, 45	sseqtrrd 3966	. . . . . . . . . . 11 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑓)
49	40, 41, 48	bnj1503 32808	. . . . . . . . . 10 ⊢ (𝜒 → (𝐹 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)))
50	49	opeq2d 4816	. . . . . . . . 9 ⊢ (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
51	50, 4	eqtr4di 2797	. . . . . . . 8 ⊢ (𝜒 → ⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = 𝑌)
52	51	fveq2d 6772	. . . . . . 7 ⊢ (𝜒 → (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) = (𝐺‘𝑌))
53	39, 52	eqtr4d 2782	. . . . . 6 ⊢ (𝜒 → (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
54	26, 53	bnj593 32704	. . . . 5 ⊢ (𝜓 → ∃𝑑(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
55	3, 4, 5, 6	bnj1519 33024	. . . . 5 ⊢ ((𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑑(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
56	54, 55	bnj1397 32793	. . . 4 ⊢ (𝜓 → (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
57	18, 56	bnj593 32704	. . 3 ⊢ (𝜑 → ∃𝑓(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
58	3, 4, 5, 6	bnj1520 33025	. . 3 ⊢ ((𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑓(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
59	57, 58	bnj1397 32793	. 2 ⊢ (𝜑 → (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
60	1, 59	bnj1459 32802	1 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1541   ∈ wcel 2109  {cab 2716  ∀wral 3065  ∃wrex 3066   ⊆ wss 3891  ⟨cop 4572  ∪ cuni 4844  ∪ ciun 4929  dom cdm 5588   ↾ cres 5590  Fun wfun 6424   Fn wfn 6425  ‘cfv 6430   predc-bnj14 32646   FrSe w-bnj15 32650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579  ax-reg 9312  ax-inf2 9360

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-om 7701  df-1o 8281  df-bnj17 32645  df-bnj14 32647  df-bnj13 32649  df-bnj15 32651  df-bnj18 32653  df-bnj19 32655

This theorem is referenced by:  bnj1500  33027