Theorem tfrlemibacc 6591

Description: Each element of 𝐵 is an acceptable function. Lemma for tfrlemi1 6597. (Contributed by Jim Kingdon, 14-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
tfrlemibacc	⊢ (𝜑 → 𝐵 ⊆ 𝐴)

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

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

Proof of Theorem tfrlemibacc

Step	Hyp	Ref	Expression
1		tfrlemi1.3	. 2 ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))}
2		simpr3 1036	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
3		tfrlemisucfn.1	. . . . . . . 8 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
4		tfrlemisucfn.2	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V))
5	4	ad2antrr 492	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V))
6		tfrlemi1.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝑥 ∈ On)
7	6	ad2antrr 492	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → 𝑥 ∈ On)
8		simplr 533	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → 𝑧 ∈ 𝑥)
9		onelon 4527	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ On)
10	7, 8, 9	syl2anc 415	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → 𝑧 ∈ On)
11		simpr1 1034	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → 𝑔 Fn 𝑧)
12		simpr2 1035	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → 𝑔 ∈ 𝐴)
13	3, 5, 10, 11, 12	tfrlemisucaccv 6590	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝐴)
14	2, 13	eqeltrd 2315	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → ℎ ∈ 𝐴)
15	14	ex 115	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑥) → ((𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})) → ℎ ∈ 𝐴))
16	15	exlimdv 1872	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑥) → (∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})) → ℎ ∈ 𝐴))
17	16	rexlimdva 2668	. . 3 ⊢ (𝜑 → (∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})) → ℎ ∈ 𝐴))
18	17	abssdv 3322	. 2 ⊢ (𝜑 → {ℎ ∣ ∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))} ⊆ 𝐴)
19	1, 18	eqsstrid 3294	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1009  ∀wal 1400   = wceq 1402  ∃wex 1545   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∃wrex 2529  Vcvv 2821   ∪ cun 3218   ⊆ wss 3220  {csn 3708  ⟨cop 3711  Oncon0 4506   ↾ cres 4774  Fun wfun 5369   Fn wfn 5370  ‘cfv 5375

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-res 4784  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383

This theorem is referenced by:  tfrlemibfn  6593  tfrlemiubacc  6595