Theorem bnj1525 33028

Description: Technical lemma for bnj1522 33031. 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
bnj1525.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1525.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1525.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1525.4	⊢ 𝐹 = ∪ 𝐶
bnj1525.5	⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)))
bnj1525.6	⊢ (𝜓 ↔ (𝜑 ∧ 𝐹 ≠ 𝐻))

Assertion

Ref	Expression
bnj1525	⊢ (𝜓 → ∀𝑥𝜓)

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

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

Proof of Theorem bnj1525

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1525.6	. . 3 ⊢ (𝜓 ↔ (𝜑 ∧ 𝐹 ≠ 𝐻))
2		bnj1525.5	. . . . 5 ⊢ (𝜑 ↔ (𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)))
3		nfv 1920	. . . . . 6 ⊢ Ⅎ𝑥 𝑅 FrSe 𝐴
4		nfv 1920	. . . . . 6 ⊢ Ⅎ𝑥 𝐻 Fn 𝐴
5		nfra1 3144	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
6	3, 4, 5	nf3an 1907	. . . . 5 ⊢ Ⅎ𝑥(𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
7	2, 6	nfxfr 1858	. . . 4 ⊢ Ⅎ𝑥𝜑
8		bnj1525.4	. . . . . 6 ⊢ 𝐹 = ∪ 𝐶
9		bnj1525.3	. . . . . . . . 9 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
10		bnj1525.1	. . . . . . . . . 10 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
11	10	bnj1309 32981	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵)
12	9, 11	bnj1307 32982	. . . . . . . 8 ⊢ (𝑤 ∈ 𝐶 → ∀𝑥 𝑤 ∈ 𝐶)
13	12	nfcii 2892	. . . . . . 7 ⊢ Ⅎ𝑥𝐶
14	13	nfuni 4851	. . . . . 6 ⊢ Ⅎ𝑥∪ 𝐶
15	8, 14	nfcxfr 2906	. . . . 5 ⊢ Ⅎ𝑥𝐹
16		nfcv 2908	. . . . 5 ⊢ Ⅎ𝑥𝐻
17	15, 16	nfne 3046	. . . 4 ⊢ Ⅎ𝑥 𝐹 ≠ 𝐻
18	7, 17	nfan 1905	. . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐹 ≠ 𝐻)
19	1, 18	nfxfr 1858	. 2 ⊢ Ⅎ𝑥𝜓
20	19	nf5ri 2191	1 ⊢ (𝜓 → ∀𝑥𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085  ∀wal 1539   = wceq 1541  {cab 2716   ≠ wne 2944  ∀wral 3065  ∃wrex 3066   ⊆ wss 3891  ⟨cop 4572  ∪ cuni 4844   ↾ cres 5590   Fn wfn 6425  ‘cfv 6430   predc-bnj14 32646   FrSe w-bnj15 32650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-uni 4845

This theorem is referenced by:  bnj1523  33030