Theorem bnj1519 34375

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
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 3281	. . . . . . . 8 ⊢ Ⅎ𝑑∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))
4	3	nfab 2908	. . . . . . 7 ⊢ Ⅎ𝑑{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
5	2, 4	nfcxfr 2900	. . . . . 6 ⊢ Ⅎ𝑑𝐶
6	5	nfuni 4915	. . . . 5 ⊢ Ⅎ𝑑∪ 𝐶
7	1, 6	nfcxfr 2900	. . . 4 ⊢ Ⅎ𝑑𝐹
8		nfcv 2902	. . . 4 ⊢ Ⅎ𝑑𝑥
9	7, 8	nffv 6901	. . 3 ⊢ Ⅎ𝑑(𝐹‘𝑥)
10		nfcv 2902	. . . 4 ⊢ Ⅎ𝑑𝐺
11		nfcv 2902	. . . . . 6 ⊢ Ⅎ𝑑 pred(𝑥, 𝐴, 𝑅)
12	7, 11	nfres 5983	. . . . 5 ⊢ Ⅎ𝑑(𝐹 ↾ pred(𝑥, 𝐴, 𝑅))
13	8, 12	nfop 4889	. . . 4 ⊢ Ⅎ𝑑⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩
14	10, 13	nffv 6901	. . 3 ⊢ Ⅎ𝑑(𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
15	9, 14	nfeq 2915	. 2 ⊢ Ⅎ𝑑(𝐹‘𝑥) = (𝐺‘⟨𝑥, (𝐹 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
16	15	nf5ri 2187	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 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