Theorem tfrlemiex 6440

Description: Lemma for tfrlemi1 6441. (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 6438	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
7		uniexg 4504	. . 3 ⊢ (𝐵 ∈ V → ∪ 𝐵 ∈ V)
8	6, 7	syl 14	. 2 ⊢ (𝜑 → ∪ 𝐵 ∈ V)
9	1, 2, 3, 4, 5	tfrlemibfn 6437	. . 3 ⊢ (𝜑 → ∪ 𝐵 Fn 𝑥)
10	1, 2, 3, 4, 5	tfrlemiubacc 6439	. . 3 ⊢ (𝜑 → ∀𝑢 ∈ 𝑥 (∪ 𝐵‘𝑢) = (𝐹‘(∪ 𝐵 ↾ 𝑢)))
11	9, 10	jca 306	. 2 ⊢ (𝜑 → (∪ 𝐵 Fn 𝑥 ∧ ∀𝑢 ∈ 𝑥 (∪ 𝐵‘𝑢) = (𝐹‘(∪ 𝐵 ↾ 𝑢))))
12		fneq1 5381	. . . 4 ⊢ (𝑓 = ∪ 𝐵 → (𝑓 Fn 𝑥 ↔ ∪ 𝐵 Fn 𝑥))
13		fveq1 5598	. . . . . 6 ⊢ (𝑓 = ∪ 𝐵 → (𝑓‘𝑢) = (∪ 𝐵‘𝑢))
14		reseq1 4972	. . . . . . 7 ⊢ (𝑓 = ∪ 𝐵 → (𝑓 ↾ 𝑢) = (∪ 𝐵 ↾ 𝑢))
15	14	fveq2d 5603	. . . . . 6 ⊢ (𝑓 = ∪ 𝐵 → (𝐹‘(𝑓 ↾ 𝑢)) = (𝐹‘(∪ 𝐵 ↾ 𝑢)))
16	13, 15	eqeq12d 2222	. . . . 5 ⊢ (𝑓 = ∪ 𝐵 → ((𝑓‘𝑢) = (𝐹‘(𝑓 ↾ 𝑢)) ↔ (∪ 𝐵‘𝑢) = (𝐹‘(∪ 𝐵 ↾ 𝑢))))
17	16	ralbidv 2508	. . . 4 ⊢ (𝑓 = ∪ 𝐵 → (∀𝑢 ∈ 𝑥 (𝑓‘𝑢) = (𝐹‘(𝑓 ↾ 𝑢)) ↔ ∀𝑢 ∈ 𝑥 (∪ 𝐵‘𝑢) = (𝐹‘(∪ 𝐵 ↾ 𝑢))))
18	12, 17	anbi12d 473	. . 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 2868	. 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 104   ∧ w3a 981  ∀wal 1371   = wceq 1373  ∃wex 1516   ∈ wcel 2178  {cab 2193  ∀wral 2486  ∃wrex 2487  Vcvv 2776   ∪ cun 3172  {csn 3643  ⟨cop 3646  ∪ cuni 3864  Oncon0 4428   ↾ cres 4695  Fun wfun 5284   Fn wfn 5285  ‘cfv 5290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-suc 4436  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-recs 6414

This theorem is referenced by:  tfrlemi1  6441