Theorem bnj1497 33339

Description: Technical lemma for bnj60 33341. 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
bnj1497.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1497.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1497.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}

Assertion

Ref	Expression
bnj1497	⊢ ∀𝑔 ∈ 𝐶 Fun 𝑔

Distinct variable groups:   𝐶,𝑔   𝑓,𝑑   𝑓,𝑔

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

Proof of Theorem bnj1497

Step	Hyp	Ref	Expression
1		bnj1497.3	. . . . . 6 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
2	1	bnj1317 33100	. . . . 5 ⊢ (𝑔 ∈ 𝐶 → ∀𝑓 𝑔 ∈ 𝐶)
3	2	nf5i 2141	. . . 4 ⊢ Ⅎ𝑓 𝑔 ∈ 𝐶
4		nfv 1916	. . . 4 ⊢ Ⅎ𝑓Fun 𝑔
5	3, 4	nfim 1898	. . 3 ⊢ Ⅎ𝑓(𝑔 ∈ 𝐶 → Fun 𝑔)
6		eleq1w 2819	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐶 ↔ 𝑔 ∈ 𝐶))
7		funeq 6505	. . . 4 ⊢ (𝑓 = 𝑔 → (Fun 𝑓 ↔ Fun 𝑔))
8	6, 7	imbi12d 344	. . 3 ⊢ (𝑓 = 𝑔 → ((𝑓 ∈ 𝐶 → Fun 𝑓) ↔ (𝑔 ∈ 𝐶 → Fun 𝑔)))
9	1	bnj1436 33118	. . . . . 6 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
10	9	bnj1299 33097	. . . . 5 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 𝑓 Fn 𝑑)
11		fnfun 6586	. . . . 5 ⊢ (𝑓 Fn 𝑑 → Fun 𝑓)
12	10, 11	bnj31 32998	. . . 4 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 Fun 𝑓)
13	12	bnj1265 33091	. . 3 ⊢ (𝑓 ∈ 𝐶 → Fun 𝑓)
14	5, 8, 13	chvarfv 2232	. 2 ⊢ (𝑔 ∈ 𝐶 → Fun 𝑔)
15	14	rgen 3063	1 ⊢ ∀𝑔 ∈ 𝐶 Fun 𝑔

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  {cab 2713  ∀wral 3061  ∃wrex 3070   ⊆ wss 3898  ⟨cop 4580   ↾ cres 5623  Fun wfun 6474   Fn wfn 6475  ‘cfv 6480   predc-bnj14 32967

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-12 2170  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-v 3443  df-in 3905  df-ss 3915  df-br 5094  df-opab 5156  df-rel 5628  df-cnv 5629  df-co 5630  df-fun 6482  df-fn 6483

This theorem is referenced by:  bnj60  33341