Theorem bnj1018g 34975

Description: Version of bnj1018 34976 with less disjoint variable conditions, but requiring ax-13 2372. (Contributed by GG, 27-Mar-2024.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1018.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1018.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1018.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj1018.4	⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
bnj1018.5	⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
bnj1018.7	⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑)
bnj1018.8	⊢ (𝜓′ ↔ [𝑝 / 𝑛]𝜓)
bnj1018.9	⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒)
bnj1018.10	⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′)
bnj1018.11	⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓′)
bnj1018.12	⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′)
bnj1018.13	⊢ 𝐷 = (ω ∖ {∅})
bnj1018.14	⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
bnj1018.15	⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
bnj1018.16	⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
bnj1018.26	⊢ (𝜒″ ↔ (𝑝 ∈ 𝐷 ∧ 𝐺 Fn 𝑝 ∧ 𝜑″ ∧ 𝜓″))
bnj1018.29	⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝜒″)
bnj1018.30	⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝))

Assertion

Ref	Expression
bnj1018g	⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))

Distinct variable groups:   𝐴,𝑓,𝑖,𝑚,𝑛,𝑦   𝐷,𝑓,𝑖,𝑛   𝑖,𝐺,𝑝   𝑅,𝑓,𝑖,𝑚,𝑛,𝑦   𝑓,𝑋,𝑖,𝑛,𝑦   𝜒,𝑝   𝜂,𝑝   𝑓,𝑝   𝜑,𝑖   𝜃,𝑝

Allowed substitution hints:   𝜑(𝑦,𝑧,𝑓,𝑚,𝑛,𝑝)   𝜓(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛)   𝜃(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛)   𝜏(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜂(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛)   𝐴(𝑧,𝑝)   𝐵(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐶(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑧,𝑚,𝑝)   𝑅(𝑧,𝑝)   𝐺(𝑦,𝑧,𝑓,𝑚,𝑛)   𝑋(𝑧,𝑚,𝑝)   𝜑′(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓′(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒′(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜑″(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓″(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒″(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj1018g

Step	Hyp	Ref	Expression
1		df-bnj17 34699	. . 3 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) ↔ ((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ ∃𝑝𝜏))
2		bnj258 34720	. . . . . . . 8 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) ↔ ((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ 𝜏))
3		bnj1018.29	. . . . . . . 8 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝜒″)
4	2, 3	sylbir 235	. . . . . . 7 ⊢ (((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ 𝜏) → 𝜒″)
5	4	ex 412	. . . . . 6 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂) → (𝜏 → 𝜒″))
6	5	eximdv 1918	. . . . 5 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂) → (∃𝑝𝜏 → ∃𝑝𝜒″))
7		bnj1018.3	. . . . . 6 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
8		bnj1018.9	. . . . . 6 ⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒)
9		bnj1018.12	. . . . . 6 ⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′)
10		bnj1018.14	. . . . . 6 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
11		bnj1018.16	. . . . . 6 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
12	7, 8, 9, 10, 11	bnj985 34966	. . . . 5 ⊢ (𝐺 ∈ 𝐵 ↔ ∃𝑝𝜒″)
13	6, 12	imbitrrdi 252	. . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂) → (∃𝑝𝜏 → 𝐺 ∈ 𝐵))
14	13	imp 406	. . 3 ⊢ (((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ ∃𝑝𝜏) → 𝐺 ∈ 𝐵)
15	1, 14	sylbi 217	. 2 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → 𝐺 ∈ 𝐵)
16		bnj1019 34791	. . 3 ⊢ (∃𝑝(𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) ↔ (𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏))
17		bnj1018.30	. . . . . 6 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝))
18	17	simp3d 1144	. . . . 5 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → suc 𝑖 ∈ 𝑝)
19		bnj1018.26	. . . . . . 7 ⊢ (𝜒″ ↔ (𝑝 ∈ 𝐷 ∧ 𝐺 Fn 𝑝 ∧ 𝜑″ ∧ 𝜓″))
20	19	bnj1235 34816	. . . . . 6 ⊢ (𝜒″ → 𝐺 Fn 𝑝)
21		fndm 6584	. . . . . 6 ⊢ (𝐺 Fn 𝑝 → dom 𝐺 = 𝑝)
22	3, 20, 21	3syl 18	. . . . 5 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → dom 𝐺 = 𝑝)
23	18, 22	eleqtrrd 2834	. . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → suc 𝑖 ∈ dom 𝐺)
24	23	exlimiv 1931	. . 3 ⊢ (∃𝑝(𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → suc 𝑖 ∈ dom 𝐺)
25	16, 24	sylbir 235	. 2 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → suc 𝑖 ∈ dom 𝐺)
26		bnj1018.1	. . 3 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
27		bnj1018.2	. . 3 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
28		bnj1018.13	. . 3 ⊢ 𝐷 = (ω ∖ {∅})
29	11	bnj918 34778	. . 3 ⊢ 𝐺 ∈ V
30		vex 3440	. . . 4 ⊢ 𝑖 ∈ V
31	30	sucex 7739	. . 3 ⊢ suc 𝑖 ∈ V
32	26, 27, 28, 10, 29, 31	bnj1015 34974	. 2 ⊢ ((𝐺 ∈ 𝐵 ∧ suc 𝑖 ∈ dom 𝐺) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
33	15, 25, 32	syl2anc 584	1 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709  ∀wral 3047  ∃wrex 3056  Vcvv 3436  [wsbc 3736   ∖ cdif 3894   ∪ cun 3895   ⊆ wss 3897  ∅c0 4280  {csn 4573  ⟨cop 4579  ∪ ciun 4939  dom cdm 5614  suc csuc 6308   Fn wfn 6476  ‘cfv 6481  ωcom 7796   ∧ w-bnj17 34698   predc-bnj14 34700   FrSe w-bnj15 34704   trClc-bnj18 34706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-13 2372  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-dm 5624  df-suc 6312  df-iota 6437  df-fn 6484  df-fv 6489  df-bnj17 34699  df-bnj18 34707

This theorem is referenced by: (None)