Theorem bnj1497 32327

Description: Technical lemma for bnj60 32329. 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 32088	. . . . 5 ⊢ (𝑔 ∈ 𝐶 → ∀𝑓 𝑔 ∈ 𝐶)
3	2	nf5i 2146	. . . 4 ⊢ Ⅎ𝑓 𝑔 ∈ 𝐶
4		nfv 1911	. . . 4 ⊢ Ⅎ𝑓Fun 𝑔
5	3, 4	nfim 1893	. . 3 ⊢ Ⅎ𝑓(𝑔 ∈ 𝐶 → Fun 𝑔)
6		eleq1w 2895	. . . 4 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐶 ↔ 𝑔 ∈ 𝐶))
7		funeq 6369	. . . 4 ⊢ (𝑓 = 𝑔 → (Fun 𝑓 ↔ Fun 𝑔))
8	6, 7	imbi12d 347	. . 3 ⊢ (𝑓 = 𝑔 → ((𝑓 ∈ 𝐶 → Fun 𝑓) ↔ (𝑔 ∈ 𝐶 → Fun 𝑔)))
9	1	bnj1436 32106	. . . . . 6 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
10	9	bnj1299 32085	. . . . 5 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 𝑓 Fn 𝑑)
11		fnfun 6447	. . . . 5 ⊢ (𝑓 Fn 𝑑 → Fun 𝑓)
12	10, 11	bnj31 31984	. . . 4 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 Fun 𝑓)
13	12	bnj1265 32079	. . 3 ⊢ (𝑓 ∈ 𝐶 → Fun 𝑓)
14	5, 8, 13	chvarfv 2238	. 2 ⊢ (𝑔 ∈ 𝐶 → Fun 𝑔)
15	14	rgen 3148	1 ⊢ ∀𝑔 ∈ 𝐶 Fun 𝑔

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  {cab 2799  ∀wral 3138  ∃wrex 3139   ⊆ wss 3935  ⟨cop 4566   ↾ cres 5551  Fun wfun 6343   Fn wfn 6344  ‘cfv 6349   predc-bnj14 31953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-in 3942  df-ss 3951  df-br 5059  df-opab 5121  df-rel 5556  df-cnv 5557  df-co 5558  df-fun 6351  df-fn 6352

This theorem is referenced by:  bnj60  32329