Theorem bnj1525 35083

Description: Technical lemma for bnj1522 35086. 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 1914	. . . . . 6 ⊢ Ⅎ𝑥 𝑅 FrSe 𝐴
4		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑥 𝐻 Fn 𝐴
5		nfra1 3284	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
6	3, 4, 5	nf3an 1901	. . . . 5 ⊢ Ⅎ𝑥(𝑅 FrSe 𝐴 ∧ 𝐻 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐺‘⟨𝑥, (𝐻 ↾ pred(𝑥, 𝐴, 𝑅))⟩))
7	2, 6	nfxfr 1853	. . . 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 35036	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐵 → ∀𝑥 𝑤 ∈ 𝐵)
12	9, 11	bnj1307 35037	. . . . . . . 8 ⊢ (𝑤 ∈ 𝐶 → ∀𝑥 𝑤 ∈ 𝐶)
13	12	nfcii 2894	. . . . . . 7 ⊢ Ⅎ𝑥𝐶
14	13	nfuni 4914	. . . . . 6 ⊢ Ⅎ𝑥∪ 𝐶
15	8, 14	nfcxfr 2903	. . . . 5 ⊢ Ⅎ𝑥𝐹
16		nfcv 2905	. . . . 5 ⊢ Ⅎ𝑥𝐻
17	15, 16	nfne 3043	. . . 4 ⊢ Ⅎ𝑥 𝐹 ≠ 𝐻
18	7, 17	nfan 1899	. . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝐹 ≠ 𝐻)
19	1, 18	nfxfr 1853	. 2 ⊢ Ⅎ𝑥𝜓
20	19	nf5ri 2195	1 ⊢ (𝜓 → ∀𝑥𝜓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1538   = wceq 1540  {cab 2714   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   ⊆ wss 3951  ⟨cop 4632  ∪ cuni 4907   ↾ cres 5687   Fn wfn 6556  ‘cfv 6561   predc-bnj14 34702   FrSe w-bnj15 34706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-uni 4908

This theorem is referenced by:  bnj1523  35085