Theorem bnj1520 35562

Description: Technical lemma for bnj1500 35564. 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
bnj1520.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1520.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1520.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1520.4	⊢ 𝐹 = ∪ 𝐶

Assertion

Ref	Expression
bnj1520	⊢ ((𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑓(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))

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

Allowed substitution hints:   𝐴(𝑥, 𝑑)   𝐵(𝑥, 𝑓, 𝑑)   𝐶(𝑥, 𝑓, 𝑑)   𝑅(𝑥, 𝑑)   𝐹(𝑥, 𝑓, 𝑑)   𝐺(𝑥, 𝑑)   𝑌(𝑥, 𝑓, 𝑑)

Proof of Theorem bnj1520

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1520.4	. . . . 5 ⊢ 𝐹 = ∪ 𝐶
2		bnj1520.3	. . . . . . . 8 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
3	2	bnj1317 35317	. . . . . . 7 ⊢ (𝑤 ∈ 𝐶 → ∀𝑓 𝑤 ∈ 𝐶)
4	3	nfcii 2913	. . . . . 6 ⊢ Ⅎ𝑓𝐶
5	4	nfuni 4877	. . . . 5 ⊢ Ⅎ𝑓∪ 𝐶
6	1, 5	nfcxfr 2922	. . . 4 ⊢ Ⅎ𝑓𝐹
7		nfcv 2924	. . . 4 ⊢ Ⅎ𝑓𝑥
8	6, 7	nffv 6892	. . 3 ⊢ Ⅎ𝑓(𝐹‘𝑥)
9		nfcv 2924	. . . 4 ⊢ Ⅎ𝑓𝐺
10		nfcv 2924	. . . . . 6 ⊢ Ⅎ𝑓 pred(𝑥, 𝐴, 𝑅)
11	6, 10	nfres 5978	. . . . 5 ⊢ Ⅎ𝑓(𝐹 ↾ pred(𝑥, 𝐴, 𝑅))
12	7, 11	nfop 4852	. . . 4 ⊢ Ⅎ𝑓⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩
13	9, 12	nffv 6892	. . 3 ⊢ Ⅎ𝑓(𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
14	8, 13	nfeq 2937	. 2 ⊢ Ⅎ𝑓(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
15	14	nf5ri 2233	1 ⊢ ((𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑓(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570  {cab 2740  ∀wral 3078  ∃wrex 3088   ⊆ wss 3902  ⟨cop 4593  ∪ cuni 4870   ↾ cres 5661   Fn wfn 6532  ‘cfv 6537   predc-bnj14 35185

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-xp 5665  df-res 5671  df-iota 6493  df-fv 6545

This theorem is used by:  bnj1501  35563