Theorem bnj1447 31114

Description: Technical lemma for bnj60 31130. 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
bnj1447.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1447.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1447.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1447.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1447.5	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1447.6	⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
bnj1447.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
bnj1447.8	⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
bnj1447.9	⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1447.10	⊢ 𝑃 = ∪ 𝐻
bnj1447.11	⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1447.12	⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
bnj1447.13	⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩

Assertion

Ref	Expression
bnj1447	⊢ ((𝑄‘𝑧) = (𝐺‘𝑊) → ∀𝑦(𝑄‘𝑧) = (𝐺‘𝑊))

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

Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐴(𝑥,𝑧,𝑓,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,𝑧,𝑓,𝑑)   𝐺(𝑥,𝑧,𝑓,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 bnj1447

Step	Hyp	Ref	Expression
1		bnj1447.12	. . . . 5 ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
2		bnj1447.10	. . . . . . 7 ⊢ 𝑃 = ∪ 𝐻
3		bnj1447.9	. . . . . . . . 9 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
4		nfre1 3005	. . . . . . . . . 10 ⊢ Ⅎ𝑦∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′
5	4	nfab 2769	. . . . . . . . 9 ⊢ Ⅎ𝑦{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
6	3, 5	nfcxfr 2762	. . . . . . . 8 ⊢ Ⅎ𝑦𝐻
7	6	nfuni 4442	. . . . . . 7 ⊢ Ⅎ𝑦∪ 𝐻
8	2, 7	nfcxfr 2762	. . . . . 6 ⊢ Ⅎ𝑦𝑃
9		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑦𝑥
10		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐺
11		bnj1447.11	. . . . . . . . . 10 ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
12		nfcv 2764	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦 pred(𝑥, 𝐴, 𝑅)
13	8, 12	nfres 5398	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝑃 ↾ pred(𝑥, 𝐴, 𝑅))
14	9, 13	nfop 4418	. . . . . . . . . 10 ⊢ Ⅎ𝑦⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
15	11, 14	nfcxfr 2762	. . . . . . . . 9 ⊢ Ⅎ𝑦𝑍
16	10, 15	nffv 6198	. . . . . . . 8 ⊢ Ⅎ𝑦(𝐺‘𝑍)
17	9, 16	nfop 4418	. . . . . . 7 ⊢ Ⅎ𝑦⟨𝑥, (𝐺‘𝑍)⟩
18	17	nfsn 4242	. . . . . 6 ⊢ Ⅎ𝑦{⟨𝑥, (𝐺‘𝑍)⟩}
19	8, 18	nfun 3769	. . . . 5 ⊢ Ⅎ𝑦(𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
20	1, 19	nfcxfr 2762	. . . 4 ⊢ Ⅎ𝑦𝑄
21		nfcv 2764	. . . 4 ⊢ Ⅎ𝑦𝑧
22	20, 21	nffv 6198	. . 3 ⊢ Ⅎ𝑦(𝑄‘𝑧)
23		bnj1447.13	. . . . 5 ⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
24		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑦 pred(𝑧, 𝐴, 𝑅)
25	20, 24	nfres 5398	. . . . . 6 ⊢ Ⅎ𝑦(𝑄 ↾ pred(𝑧, 𝐴, 𝑅))
26	21, 25	nfop 4418	. . . . 5 ⊢ Ⅎ𝑦⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
27	23, 26	nfcxfr 2762	. . . 4 ⊢ Ⅎ𝑦𝑊
28	10, 27	nffv 6198	. . 3 ⊢ Ⅎ𝑦(𝐺‘𝑊)
29	22, 28	nfeq 2776	. 2 ⊢ Ⅎ𝑦(𝑄‘𝑧) = (𝐺‘𝑊)
30	29	nf5ri 2065	1 ⊢ ((𝑄‘𝑧) = (𝐺‘𝑊) → ∀𝑦(𝑄‘𝑧) = (𝐺‘𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916  [wsbc 3435   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  {csn 4177  ⟨cop 4183  ∪ cuni 4436   class class class wbr 4653  dom cdm 5114   ↾ cres 5116   Fn wfn 5883  ‘cfv 5888   predc-bnj14 30754   FrSe w-bnj15 30758   trClc-bnj18 30760

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-res 5126  df-iota 5851  df-fv 5896

This theorem is referenced by:  bnj1450  31118