Theorem bnj1519 35362

Description: Technical lemma for bnj1500 35365. 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
bnj1519.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1519.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1519.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1519.4	⊢ 𝐹 = ∪ 𝐶

Assertion

Ref	Expression
bnj1519	⊢ ((𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑑(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))

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

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

Proof of Theorem bnj1519

Step	Hyp	Ref	Expression
1		bnj1519.4	. . . . 5 ⊢ 𝐹 = ∪ 𝐶
2		bnj1519.3	. . . . . . 7 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
3		nfre1 3289	. . . . . . . 8 ⊢ Ⅎ𝑑∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))
4	3	nfab 2932	. . . . . . 7 ⊢ Ⅎ𝑑{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
5	2, 4	nfcxfr 2924	. . . . . 6 ⊢ Ⅎ𝑑𝐶
6	5	nfuni 4874	. . . . 5 ⊢ Ⅎ𝑑∪ 𝐶
7	1, 6	nfcxfr 2924	. . . 4 ⊢ Ⅎ𝑑𝐹
8		nfcv 2926	. . . 4 ⊢ Ⅎ𝑑𝑥
9	7, 8	nffv 6879	. . 3 ⊢ Ⅎ𝑑(𝐹‘𝑥)
10		nfcv 2926	. . . 4 ⊢ Ⅎ𝑑𝐺
11		nfcv 2926	. . . . . 6 ⊢ Ⅎ𝑑 pred(𝑥, 𝐴, 𝑅)
12	7, 11	nfres 5969	. . . . 5 ⊢ Ⅎ𝑑(𝐹 ↾ pred(𝑥, 𝐴, 𝑅))
13	8, 12	nfop 4849	. . . 4 ⊢ Ⅎ𝑑⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩
14	10, 13	nffv 6879	. . 3 ⊢ Ⅎ𝑑(𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
15	9, 14	nfeq 2939	. 2 ⊢ Ⅎ𝑑(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
16	15	nf5ri 2232	1 ⊢ ((𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩) → ∀𝑑(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩))

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1560   = wceq 1562  {cab 2742  ∀wral 3078  ∃wrex 3088   ⊆ wss 3906  ⟨cop 4590  ∪ cuni 4867   ↾ cres 5651   Fn wfn 6518  ‘cfv 6523   predc-bnj14 34986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-xp 5655  df-res 5661  df-iota 6479  df-fv 6531

This theorem is referenced by:  bnj1501  35364