Theorem bnj66 35257

Description: Technical lemma for bnj60 35459. 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
bnj66.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj66.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj66.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}

Assertion

Ref	Expression
bnj66	⊢ (𝑔 ∈ 𝐶 → Rel 𝑔)

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓,𝑔   𝑓,𝐺,𝑔   𝑅,𝑓   𝑔,𝑌   𝑓,𝑑,𝑔   𝑥,𝑓,𝑔

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

Proof of Theorem bnj66

Step	Hyp	Ref	Expression
1		bnj66.3	. . . 4 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
2		fneq1 6626	. . . . . . 7 ⊢ (𝑔 = 𝑓 → (𝑔 Fn 𝑑 ↔ 𝑓 Fn 𝑑))
3		fveq1 6880	. . . . . . . . 9 ⊢ (𝑔 = 𝑓 → (𝑔‘𝑥) = (𝑓‘𝑥))
4		reseq1 5971	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)))
5	4	opeq2d 4844	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
6		bnj66.2	. . . . . . . . . . 11 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
7	5, 6	eqtr4di 2815	. . . . . . . . . 10 ⊢ (𝑔 = 𝑓 → ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = 𝑌)
8	7	fveq2d 6885	. . . . . . . . 9 ⊢ (𝑔 = 𝑓 → (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩) = (𝐺‘𝑌))
9	3, 8	eqeq12d 2778	. . . . . . . 8 ⊢ (𝑔 = 𝑓 → ((𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩) ↔ (𝑓‘𝑥) = (𝐺‘𝑌)))
10	9	ralbidv 3187	. . . . . . 7 ⊢ (𝑔 = 𝑓 → (∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩) ↔ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
11	2, 10	anbi12d 643	. . . . . 6 ⊢ (𝑔 = 𝑓 → ((𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) ↔ (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))))
12	11	rexbidv 3188	. . . . 5 ⊢ (𝑔 = 𝑓 → (∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) ↔ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))))
13	12	cbvabv 2832	. . . 4 ⊢ {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))} = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
14	1, 13	eqtr4i 2788	. . 3 ⊢ 𝐶 = {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))}
15	14	bnj1436 35236	. 2 ⊢ (𝑔 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩)))
16		bnj1239 35202	. 2 ⊢ (∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) → ∃𝑑 ∈ 𝐵 𝑔 Fn 𝑑)
17		fnrel 6637	. . 3 ⊢ (𝑔 Fn 𝑑 → Rel 𝑔)
18	17	rexlimivw 3161	. 2 ⊢ (∃𝑑 ∈ 𝐵 𝑔 Fn 𝑑 → Rel 𝑔)
19	15, 16, 18	3syl 19	1 ⊢ (𝑔 ∈ 𝐶 → Rel 𝑔)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  {cab 2740  ∀wral 3078  ∃wrex 3088   ⊆ wss 3904  ⟨cop 4594   ↾ cres 5662  Rel wrel 5665   Fn wfn 6531  ‘cfv 6536   predc-bnj14 35086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-res 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544

This theorem is used by:  bnj1321  35424