Theorem bnj1448 32694

Description: Technical lemma for bnj60 32709. 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
bnj1448.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1448.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1448.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1448.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1448.5	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1448.6	⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
bnj1448.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
bnj1448.8	⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
bnj1448.9	⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1448.10	⊢ 𝑃 = ∪ 𝐻
bnj1448.11	⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1448.12	⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
bnj1448.13	⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩

Assertion

Ref	Expression
bnj1448	⊢ ((𝑄‘𝑧) = (𝐺‘𝑊) → ∀𝑓(𝑄‘𝑧) = (𝐺‘𝑊))

Distinct variable groups:   𝐴,𝑓   𝑓,𝐺   𝑅,𝑓   𝑥,𝑓   𝑧,𝑓

Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐴(𝑥,𝑦,𝑧,𝑑)   𝐵(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐶(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐷(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑄(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑅(𝑥,𝑦,𝑧,𝑑)   𝐺(𝑥,𝑦,𝑧,𝑑)   𝐻(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑊(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑌(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑍(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏′(𝑥,𝑦,𝑧,𝑓,𝑑)

Proof of Theorem bnj1448

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1448.12	. . . . 5 ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
2		bnj1448.10	. . . . . . 7 ⊢ 𝑃 = ∪ 𝐻
3		bnj1448.9	. . . . . . . . . 10 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
4	3	bnj1317 32468	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐻 → ∀𝑓 𝑤 ∈ 𝐻)
5	4	nfcii 2881	. . . . . . . 8 ⊢ Ⅎ𝑓𝐻
6	5	nfuni 4812	. . . . . . 7 ⊢ Ⅎ𝑓∪ 𝐻
7	2, 6	nfcxfr 2895	. . . . . 6 ⊢ Ⅎ𝑓𝑃
8		nfcv 2897	. . . . . . . 8 ⊢ Ⅎ𝑓𝑥
9		nfcv 2897	. . . . . . . . 9 ⊢ Ⅎ𝑓𝐺
10		bnj1448.11	. . . . . . . . . 10 ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
11		nfcv 2897	. . . . . . . . . . . 12 ⊢ Ⅎ𝑓 pred(𝑥, 𝐴, 𝑅)
12	7, 11	nfres 5838	. . . . . . . . . . 11 ⊢ Ⅎ𝑓(𝑃 ↾ pred(𝑥, 𝐴, 𝑅))
13	8, 12	nfop 4786	. . . . . . . . . 10 ⊢ Ⅎ𝑓⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
14	10, 13	nfcxfr 2895	. . . . . . . . 9 ⊢ Ⅎ𝑓𝑍
15	9, 14	nffv 6705	. . . . . . . 8 ⊢ Ⅎ𝑓(𝐺‘𝑍)
16	8, 15	nfop 4786	. . . . . . 7 ⊢ Ⅎ𝑓⟨𝑥, (𝐺‘𝑍)⟩
17	16	nfsn 4609	. . . . . 6 ⊢ Ⅎ𝑓{⟨𝑥, (𝐺‘𝑍)⟩}
18	7, 17	nfun 4065	. . . . 5 ⊢ Ⅎ𝑓(𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
19	1, 18	nfcxfr 2895	. . . 4 ⊢ Ⅎ𝑓𝑄
20		nfcv 2897	. . . 4 ⊢ Ⅎ𝑓𝑧
21	19, 20	nffv 6705	. . 3 ⊢ Ⅎ𝑓(𝑄‘𝑧)
22		bnj1448.13	. . . . 5 ⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
23		nfcv 2897	. . . . . . 7 ⊢ Ⅎ𝑓 pred(𝑧, 𝐴, 𝑅)
24	19, 23	nfres 5838	. . . . . 6 ⊢ Ⅎ𝑓(𝑄 ↾ pred(𝑧, 𝐴, 𝑅))
25	20, 24	nfop 4786	. . . . 5 ⊢ Ⅎ𝑓⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
26	22, 25	nfcxfr 2895	. . . 4 ⊢ Ⅎ𝑓𝑊
27	9, 26	nffv 6705	. . 3 ⊢ Ⅎ𝑓(𝐺‘𝑊)
28	21, 27	nfeq 2910	. 2 ⊢ Ⅎ𝑓(𝑄‘𝑧) = (𝐺‘𝑊)
29	28	nf5ri 2195	1 ⊢ ((𝑄‘𝑧) = (𝐺‘𝑊) → ∀𝑓(𝑄‘𝑧) = (𝐺‘𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089  ∀wal 1541   = wceq 1543  ∃wex 1787   ∈ wcel 2112  {cab 2714   ≠ wne 2932  ∀wral 3051  ∃wrex 3052  {crab 3055  [wsbc 3683   ∪ cun 3851   ⊆ wss 3853  ∅c0 4223  {csn 4527  ⟨cop 4533  ∪ cuni 4805   class class class wbr 5039  dom cdm 5536   ↾ cres 5538   Fn wfn 6353  ‘cfv 6358   predc-bnj14 32333   FrSe w-bnj15 32337   trClc-bnj18 32339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-xp 5542  df-res 5548  df-iota 6316  df-fv 6366

This theorem is referenced by:  bnj1450  32697  bnj1463  32702