Theorem tfrlemibacc 6384

Description: Each element of 𝐵 is an acceptable function. Lemma for tfrlemi1 6390. (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 1007	. . . . . . 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 488	. . . . . . . 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 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → 𝑥 ∈ On)
8		simplr 528	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → 𝑧 ∈ 𝑥)
9		onelon 4419	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ On)
10	7, 8, 9	syl2anc 411	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → 𝑧 ∈ On)
11		simpr1 1005	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → 𝑔 Fn 𝑧)
12		simpr2 1006	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → 𝑔 ∈ 𝐴)
13	3, 5, 10, 11, 12	tfrlemisucaccv 6383	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝐴)
14	2, 13	eqeltrd 2273	. . . . . 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 1833	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑥) → (∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})) → ℎ ∈ 𝐴))
17	16	rexlimdva 2614	. . 3 ⊢ (𝜑 → (∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})) → ℎ ∈ 𝐴))
18	17	abssdv 3257	. 2 ⊢ (𝜑 → {ℎ ∣ ∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))} ⊆ 𝐴)
19	1, 18	eqsstrid 3229	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980  ∀wal 1362   = wceq 1364  ∃wex 1506   ∈ wcel 2167  {cab 2182  ∀wral 2475  ∃wrex 2476  Vcvv 2763   ∪ cun 3155   ⊆ wss 3157  {csn 3622  ⟨cop 3625  Oncon0 4398   ↾ cres 4665  Fun wfun 5252   Fn wfn 5253  ‘cfv 5258

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266

This theorem is referenced by:  tfrlemibfn  6386  tfrlemiubacc  6388