Theorem bnj1520 34376

Description: Technical lemma for bnj1500 34378. 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 34131	. . . . . . 7 ⊢ (𝑤 ∈ 𝐶 → ∀𝑓 𝑤 ∈ 𝐶)
4	3	nfcii 2886	. . . . . 6 ⊢ Ⅎ𝑓𝐶
5	4	nfuni 4915	. . . . 5 ⊢ Ⅎ𝑓∪ 𝐶
6	1, 5	nfcxfr 2900	. . . 4 ⊢ Ⅎ𝑓𝐹
7		nfcv 2902	. . . 4 ⊢ Ⅎ𝑓𝑥
8	6, 7	nffv 6901	. . 3 ⊢ Ⅎ𝑓(𝐹‘𝑥)
9		nfcv 2902	. . . 4 ⊢ Ⅎ𝑓𝐺
10		nfcv 2902	. . . . . 6 ⊢ Ⅎ𝑓 pred(𝑥, 𝐴, 𝑅)
11	6, 10	nfres 5983	. . . . 5 ⊢ Ⅎ𝑓(𝐹 ↾ pred(𝑥, 𝐴, 𝑅))
12	7, 11	nfop 4889	. . . 4 ⊢ Ⅎ𝑓⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩
13	9, 12	nffv 6901	. . 3 ⊢ Ⅎ𝑓(𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
14	8, 13	nfeq 2915	. 2 ⊢ Ⅎ𝑓(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
15	14	nf5ri 2187	1 ⊢ ((𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑓(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540  {cab 2708  ∀wral 3060  ∃wrex 3069   ⊆ wss 3948  ⟨cop 4634  ∪ cuni 4908   ↾ cres 5678   Fn wfn 6538  ‘cfv 6543   predc-bnj14 33998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-res 5688  df-iota 6495  df-fv 6551

This theorem is referenced by:  bnj1501  34377