Theorem tfrlemiex 6228

Description: Lemma for tfrlemi1 6229. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)

Hypotheses

Ref	Expression
tfrlemisucfn.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
tfrlemisucfn.2	⊢ (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V))
tfrlemi1.3	⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))}
tfrlemi1.4	⊢ (𝜑 → 𝑥 ∈ On)
tfrlemi1.5	⊢ (𝜑 → ∀𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))))

Assertion

Ref	Expression
tfrlemiex	⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑢 ∈ 𝑥 (𝑓‘𝑢) = (𝐹‘(𝑓 ↾ 𝑢))))

Distinct variable groups:   𝑓,𝑔,ℎ,𝑢,𝑤,𝑥,𝑦,𝑧,𝐴   𝑓,𝐹,𝑔,ℎ,𝑢,𝑤,𝑥,𝑦,𝑧   𝜑,𝑤,𝑦   𝑢,𝐵,𝑤,𝑓,𝑔,ℎ,𝑧   𝜑,𝑔,ℎ,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑢,𝑓)   𝐵(𝑥,𝑦)

Proof of Theorem tfrlemiex

Step	Hyp	Ref	Expression
1		tfrlemisucfn.1	. . . 4 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
2		tfrlemisucfn.2	. . . 4 ⊢ (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V))
3		tfrlemi1.3	. . . 4 ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))}
4		tfrlemi1.4	. . . 4 ⊢ (𝜑 → 𝑥 ∈ On)
5		tfrlemi1.5	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))))
6	1, 2, 3, 4, 5	tfrlemibex 6226	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
7		uniexg 4361	. . 3 ⊢ (𝐵 ∈ V → ∪ 𝐵 ∈ V)
8	6, 7	syl 14	. 2 ⊢ (𝜑 → ∪ 𝐵 ∈ V)
9	1, 2, 3, 4, 5	tfrlemibfn 6225	. . 3 ⊢ (𝜑 → ∪ 𝐵 Fn 𝑥)
10	1, 2, 3, 4, 5	tfrlemiubacc 6227	. . 3 ⊢ (𝜑 → ∀𝑢 ∈ 𝑥 (∪ 𝐵‘𝑢) = (𝐹‘(∪ 𝐵 ↾ 𝑢)))
11	9, 10	jca 304	. 2 ⊢ (𝜑 → (∪ 𝐵 Fn 𝑥 ∧ ∀𝑢 ∈ 𝑥 (∪ 𝐵‘𝑢) = (𝐹‘(∪ 𝐵 ↾ 𝑢))))
12		fneq1 5211	. . . 4 ⊢ (𝑓 = ∪ 𝐵 → (𝑓 Fn 𝑥 ↔ ∪ 𝐵 Fn 𝑥))
13		fveq1 5420	. . . . . 6 ⊢ (𝑓 = ∪ 𝐵 → (𝑓‘𝑢) = (∪ 𝐵‘𝑢))
14		reseq1 4813	. . . . . . 7 ⊢ (𝑓 = ∪ 𝐵 → (𝑓 ↾ 𝑢) = (∪ 𝐵 ↾ 𝑢))
15	14	fveq2d 5425	. . . . . 6 ⊢ (𝑓 = ∪ 𝐵 → (𝐹‘(𝑓 ↾ 𝑢)) = (𝐹‘(∪ 𝐵 ↾ 𝑢)))
16	13, 15	eqeq12d 2154	. . . . 5 ⊢ (𝑓 = ∪ 𝐵 → ((𝑓‘𝑢) = (𝐹‘(𝑓 ↾ 𝑢)) ↔ (∪ 𝐵‘𝑢) = (𝐹‘(∪ 𝐵 ↾ 𝑢))))
17	16	ralbidv 2437	. . . 4 ⊢ (𝑓 = ∪ 𝐵 → (∀𝑢 ∈ 𝑥 (𝑓‘𝑢) = (𝐹‘(𝑓 ↾ 𝑢)) ↔ ∀𝑢 ∈ 𝑥 (∪ 𝐵‘𝑢) = (𝐹‘(∪ 𝐵 ↾ 𝑢))))
18	12, 17	anbi12d 464	. . 3 ⊢ (𝑓 = ∪ 𝐵 → ((𝑓 Fn 𝑥 ∧ ∀𝑢 ∈ 𝑥 (𝑓‘𝑢) = (𝐹‘(𝑓 ↾ 𝑢))) ↔ (∪ 𝐵 Fn 𝑥 ∧ ∀𝑢 ∈ 𝑥 (∪ 𝐵‘𝑢) = (𝐹‘(∪ 𝐵 ↾ 𝑢)))))
19	18	spcegv 2774	. 2 ⊢ (∪ 𝐵 ∈ V → ((∪ 𝐵 Fn 𝑥 ∧ ∀𝑢 ∈ 𝑥 (∪ 𝐵‘𝑢) = (𝐹‘(∪ 𝐵 ↾ 𝑢))) → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑢 ∈ 𝑥 (𝑓‘𝑢) = (𝐹‘(𝑓 ↾ 𝑢)))))
20	8, 11, 19	sylc 62	1 ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝑥 ∧ ∀𝑢 ∈ 𝑥 (𝑓‘𝑢) = (𝐹‘(𝑓 ↾ 𝑢))))

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 962  ∀wal 1329   = wceq 1331  ∃wex 1468   ∈ wcel 1480  {cab 2125  ∀wral 2416  ∃wrex 2417  Vcvv 2686   ∪ cun 3069  {csn 3527  ⟨cop 3530  ∪ cuni 3736  Oncon0 4285   ↾ cres 4541  Fun wfun 5117   Fn wfn 5118  ‘cfv 5123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-recs 6202

This theorem is referenced by:  tfrlemi1  6229