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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1018g | Structured version Visualization version GIF version |
Description: Version of bnj1018 33633 with less disjoint variable conditions, but requiring ax-13 2371. (Contributed by Gino Giotto, 27-Mar-2024.) (New usage is discouraged.) |
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)) |
Ref | Expression |
---|---|
bnj1018g | ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bnj17 33356 | . . 3 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) ↔ ((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ ∃𝑝𝜏)) | |
2 | bnj258 33377 | . . . . . . . 8 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) ↔ ((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ 𝜏)) | |
3 | bnj1018.29 | . . . . . . . 8 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝜒″) | |
4 | 2, 3 | sylbir 234 | . . . . . . 7 ⊢ (((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ 𝜏) → 𝜒″) |
5 | 4 | ex 414 | . . . . . 6 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂) → (𝜏 → 𝜒″)) |
6 | 5 | eximdv 1921 | . . . . 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 33623 | . . . . 5 ⊢ (𝐺 ∈ 𝐵 ↔ ∃𝑝𝜒″) |
13 | 6, 12 | syl6ibr 252 | . . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂) → (∃𝑝𝜏 → 𝐺 ∈ 𝐵)) |
14 | 13 | imp 408 | . . 3 ⊢ (((𝜃 ∧ 𝜒 ∧ 𝜂) ∧ ∃𝑝𝜏) → 𝐺 ∈ 𝐵) |
15 | 1, 14 | sylbi 216 | . 2 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → 𝐺 ∈ 𝐵) |
16 | bnj1019 33448 | . . 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 1145 | . . . . 5 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → suc 𝑖 ∈ 𝑝) |
19 | bnj1018.26 | . . . . . . 7 ⊢ (𝜒″ ↔ (𝑝 ∈ 𝐷 ∧ 𝐺 Fn 𝑝 ∧ 𝜑″ ∧ 𝜓″)) | |
20 | 19 | bnj1235 33473 | . . . . . 6 ⊢ (𝜒″ → 𝐺 Fn 𝑝) |
21 | fndm 6606 | . . . . . 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 2837 | . . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → suc 𝑖 ∈ dom 𝐺) |
24 | 23 | exlimiv 1934 | . . 3 ⊢ (∃𝑝(𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → suc 𝑖 ∈ dom 𝐺) |
25 | 16, 24 | sylbir 234 | . 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 33435 | . . 3 ⊢ 𝐺 ∈ V |
30 | vex 3448 | . . . 4 ⊢ 𝑖 ∈ V | |
31 | 30 | sucex 7742 | . . 3 ⊢ suc 𝑖 ∈ V |
32 | 26, 27, 28, 10, 29, 31 | bnj1015 33631 | . 2 ⊢ ((𝐺 ∈ 𝐵 ∧ suc 𝑖 ∈ dom 𝐺) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
33 | 15, 25, 32 | syl2anc 585 | 1 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∃wex 1782 ∈ wcel 2107 {cab 2710 ∀wral 3061 ∃wrex 3070 Vcvv 3444 [wsbc 3740 ∖ cdif 3908 ∪ cun 3909 ⊆ wss 3911 ∅c0 4283 {csn 4587 ⟨cop 4593 ∪ ciun 4955 dom cdm 5634 suc csuc 6320 Fn wfn 6492 ‘cfv 6497 ωcom 7803 ∧ w-bnj17 33355 predc-bnj14 33357 FrSe w-bnj15 33361 trClc-bnj18 33363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-13 2371 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-dm 5644 df-suc 6324 df-iota 6449 df-fn 6500 df-fv 6505 df-bnj17 33356 df-bnj18 33364 |
This theorem is referenced by: (None) |
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