Theorem bnj66 31811

Description: Technical lemma for bnj60 32011. 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 6274	. . . . . . 7 ⊢ (𝑔 = 𝑓 → (𝑔 Fn 𝑑 ↔ 𝑓 Fn 𝑑))
3		fveq1 6495	. . . . . . . . 9 ⊢ (𝑔 = 𝑓 → (𝑔‘𝑥) = (𝑓‘𝑥))
4		reseq1 5686	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (𝑔 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑓 ↾ pred(𝑥, 𝐴, 𝑅)))
5	4	opeq2d 4680	. . . . . . . . . . 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	syl6eqr 2825	. . . . . . . . . 10 ⊢ (𝑔 = 𝑓 → ⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩ = 𝑌)
8	7	fveq2d 6500	. . . . . . . . 9 ⊢ (𝑔 = 𝑓 → (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩) = (𝐺‘𝑌))
9	3, 8	eqeq12d 2786	. . . . . . . 8 ⊢ (𝑔 = 𝑓 → ((𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩) ↔ (𝑓‘𝑥) = (𝐺‘𝑌)))
10	9	ralbidv 3140	. . . . . . 7 ⊢ (𝑔 = 𝑓 → (∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩) ↔ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
11	2, 10	anbi12d 622	. . . . . 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 3235	. . . . 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 2903	. . . 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 2798	. . 3 ⊢ 𝐶 = {𝑔 ∣ ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩))}
15	14	bnj1436 31791	. 2 ⊢ (𝑔 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩)))
16		bnj1239 31757	. 2 ⊢ (∃𝑑 ∈ 𝐵 (𝑔 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑔‘𝑥) = (𝐺‘⟨𝑥, (𝑔 ↾ pred(𝑥, 𝐴, 𝑅))⟩)) → ∃𝑑 ∈ 𝐵 𝑔 Fn 𝑑)
17		fnrel 6284	. . 3 ⊢ (𝑔 Fn 𝑑 → Rel 𝑔)
18	17	rexlimivw 3220	. 2 ⊢ (∃𝑑 ∈ 𝐵 𝑔 Fn 𝑑 → Rel 𝑔)
19	15, 16, 18	3syl 18	1 ⊢ (𝑔 ∈ 𝐶 → Rel 𝑔)

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1508   ∈ wcel 2051  {cab 2751  ∀wral 3081  ∃wrex 3082   ⊆ wss 3822  ⟨cop 4441   ↾ cres 5405  Rel wrel 5408   Fn wfn 6180  ‘cfv 6185   predc-bnj14 31638

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-res 5415  df-iota 6149  df-fun 6187  df-fn 6188  df-fv 6193

This theorem is referenced by:  bnj1321  31976