# Metamath Proof Explorer

## Theorem bnj908

Description: Technical lemma for bnj69 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj908.1 ${⊢}{\phi }↔{f}\left(\varnothing \right)=pred\left({x},{A},{R}\right)$
bnj908.2 ${⊢}{\psi }↔\forall {i}\in \mathrm{\omega }\phantom{\rule{.4em}{0ex}}\left(\mathrm{suc}{i}\in {n}\to {f}\left(\mathrm{suc}{i}\right)=\bigcup _{{y}\in {f}\left({i}\right)}pred\left({y},{A},{R}\right)\right)$
bnj908.3 ${⊢}{D}=\mathrm{\omega }\setminus \left\{\varnothing \right\}$
bnj908.4 ${⊢}{\chi }↔\left(\left({R}FrSe{A}\wedge {x}\in {A}\right)\to \exists !{f}\phantom{\rule{.4em}{0ex}}\left({f}Fn{n}\wedge {\phi }\wedge {\psi }\right)\right)$
bnj908.5
bnj908.10 No typesetting found for |- ( ph' <-> [. m / n ]. ph ) with typecode |-
bnj908.11 No typesetting found for |- ( ps' <-> [. m / n ]. ps ) with typecode |-
bnj908.12 No typesetting found for |- ( ch' <-> [. m / n ]. ch ) with typecode |-
bnj908.13 No typesetting found for |- ( ph" <-> [. G / f ]. ph ) with typecode |-
bnj908.14 No typesetting found for |- ( ps" <-> [. G / f ]. ps ) with typecode |-
bnj908.15 No typesetting found for |- ( ch" <-> [. G / f ]. ch ) with typecode |-
bnj908.16 ${⊢}{G}={f}\cup \left\{⟨{m},\bigcup _{{y}\in {f}\left({p}\right)}pred\left({y},{A},{R}\right)⟩\right\}$
bnj908.17 No typesetting found for |- ( ta <-> ( f Fn m /\ ph' /\ ps' ) ) with typecode |-
bnj908.18 ${⊢}{\sigma }↔\left({m}\in {D}\wedge {n}=\mathrm{suc}{m}\wedge {p}\in {m}\right)$
bnj908.19 ${⊢}{\eta }↔\left({m}\in {D}\wedge {n}=\mathrm{suc}{m}\wedge {p}\in \mathrm{\omega }\wedge {m}=\mathrm{suc}{p}\right)$
bnj908.20 ${⊢}{\zeta }↔\left({i}\in \mathrm{\omega }\wedge \mathrm{suc}{i}\in {n}\wedge {m}=\mathrm{suc}{i}\right)$
bnj908.21 ${⊢}{\rho }↔\left({i}\in \mathrm{\omega }\wedge \mathrm{suc}{i}\in {n}\wedge {m}\ne \mathrm{suc}{i}\right)$
bnj908.22 ${⊢}{B}=\bigcup _{{y}\in {f}\left({i}\right)}pred\left({y},{A},{R}\right)$
bnj908.23 ${⊢}{C}=\bigcup _{{y}\in {f}\left({p}\right)}pred\left({y},{A},{R}\right)$
bnj908.24 ${⊢}{K}=\bigcup _{{y}\in {G}\left({i}\right)}pred\left({y},{A},{R}\right)$
bnj908.25 ${⊢}{L}=\bigcup _{{y}\in {G}\left({p}\right)}pred\left({y},{A},{R}\right)$
bnj908.26 ${⊢}{G}={f}\cup \left\{⟨{m},{C}⟩\right\}$
Assertion bnj908 Could not format assertion : No typesetting found for |- ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ( G Fn n /\ ph" /\ ps" ) ) with typecode |-

### Proof

Step Hyp Ref Expression
1 bnj908.1 ${⊢}{\phi }↔{f}\left(\varnothing \right)=pred\left({x},{A},{R}\right)$
2 bnj908.2 ${⊢}{\psi }↔\forall {i}\in \mathrm{\omega }\phantom{\rule{.4em}{0ex}}\left(\mathrm{suc}{i}\in {n}\to {f}\left(\mathrm{suc}{i}\right)=\bigcup _{{y}\in {f}\left({i}\right)}pred\left({y},{A},{R}\right)\right)$
3 bnj908.3 ${⊢}{D}=\mathrm{\omega }\setminus \left\{\varnothing \right\}$
4 bnj908.4 ${⊢}{\chi }↔\left(\left({R}FrSe{A}\wedge {x}\in {A}\right)\to \exists !{f}\phantom{\rule{.4em}{0ex}}\left({f}Fn{n}\wedge {\phi }\wedge {\psi }\right)\right)$
5 bnj908.5
6 bnj908.10 Could not format ( ph' <-> [. m / n ]. ph ) : No typesetting found for |- ( ph' <-> [. m / n ]. ph ) with typecode |-
7 bnj908.11 Could not format ( ps' <-> [. m / n ]. ps ) : No typesetting found for |- ( ps' <-> [. m / n ]. ps ) with typecode |-
8 bnj908.12 Could not format ( ch' <-> [. m / n ]. ch ) : No typesetting found for |- ( ch' <-> [. m / n ]. ch ) with typecode |-
9 bnj908.13 Could not format ( ph" <-> [. G / f ]. ph ) : No typesetting found for |- ( ph" <-> [. G / f ]. ph ) with typecode |-
10 bnj908.14 Could not format ( ps" <-> [. G / f ]. ps ) : No typesetting found for |- ( ps" <-> [. G / f ]. ps ) with typecode |-
11 bnj908.15 Could not format ( ch" <-> [. G / f ]. ch ) : No typesetting found for |- ( ch" <-> [. G / f ]. ch ) with typecode |-
12 bnj908.16 ${⊢}{G}={f}\cup \left\{⟨{m},\bigcup _{{y}\in {f}\left({p}\right)}pred\left({y},{A},{R}\right)⟩\right\}$
13 bnj908.17 Could not format ( ta <-> ( f Fn m /\ ph' /\ ps' ) ) : No typesetting found for |- ( ta <-> ( f Fn m /\ ph' /\ ps' ) ) with typecode |-
14 bnj908.18 ${⊢}{\sigma }↔\left({m}\in {D}\wedge {n}=\mathrm{suc}{m}\wedge {p}\in {m}\right)$
15 bnj908.19 ${⊢}{\eta }↔\left({m}\in {D}\wedge {n}=\mathrm{suc}{m}\wedge {p}\in \mathrm{\omega }\wedge {m}=\mathrm{suc}{p}\right)$
16 bnj908.20 ${⊢}{\zeta }↔\left({i}\in \mathrm{\omega }\wedge \mathrm{suc}{i}\in {n}\wedge {m}=\mathrm{suc}{i}\right)$
17 bnj908.21 ${⊢}{\rho }↔\left({i}\in \mathrm{\omega }\wedge \mathrm{suc}{i}\in {n}\wedge {m}\ne \mathrm{suc}{i}\right)$
18 bnj908.22 ${⊢}{B}=\bigcup _{{y}\in {f}\left({i}\right)}pred\left({y},{A},{R}\right)$
19 bnj908.23 ${⊢}{C}=\bigcup _{{y}\in {f}\left({p}\right)}pred\left({y},{A},{R}\right)$
20 bnj908.24 ${⊢}{K}=\bigcup _{{y}\in {G}\left({i}\right)}pred\left({y},{A},{R}\right)$
21 bnj908.25 ${⊢}{L}=\bigcup _{{y}\in {G}\left({p}\right)}pred\left({y},{A},{R}\right)$
22 bnj908.26 ${⊢}{G}={f}\cup \left\{⟨{m},{C}⟩\right\}$
23 bnj248 Could not format ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) <-> ( ( ( R _FrSe A /\ x e. A ) /\ ch' ) /\ et ) ) : No typesetting found for |- ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) <-> ( ( ( R _FrSe A /\ x e. A ) /\ ch' ) /\ et ) ) with typecode |-
24 vex ${⊢}{m}\in \mathrm{V}$
25 4 6 7 8 24 bnj207 Could not format ( ch' <-> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn m /\ ph' /\ ps' ) ) ) : No typesetting found for |- ( ch' <-> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn m /\ ph' /\ ps' ) ) ) with typecode |-
26 25 biimpi Could not format ( ch' -> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn m /\ ph' /\ ps' ) ) ) : No typesetting found for |- ( ch' -> ( ( R _FrSe A /\ x e. A ) -> E! f ( f Fn m /\ ph' /\ ps' ) ) ) with typecode |-
27 euex Could not format ( E! f ( f Fn m /\ ph' /\ ps' ) -> E. f ( f Fn m /\ ph' /\ ps' ) ) : No typesetting found for |- ( E! f ( f Fn m /\ ph' /\ ps' ) -> E. f ( f Fn m /\ ph' /\ ps' ) ) with typecode |-
28 26 27 syl6 Could not format ( ch' -> ( ( R _FrSe A /\ x e. A ) -> E. f ( f Fn m /\ ph' /\ ps' ) ) ) : No typesetting found for |- ( ch' -> ( ( R _FrSe A /\ x e. A ) -> E. f ( f Fn m /\ ph' /\ ps' ) ) ) with typecode |-
29 28 impcom Could not format ( ( ( R _FrSe A /\ x e. A ) /\ ch' ) -> E. f ( f Fn m /\ ph' /\ ps' ) ) : No typesetting found for |- ( ( ( R _FrSe A /\ x e. A ) /\ ch' ) -> E. f ( f Fn m /\ ph' /\ ps' ) ) with typecode |-
30 29 13 bnj1198 Could not format ( ( ( R _FrSe A /\ x e. A ) /\ ch' ) -> E. f ta ) : No typesetting found for |- ( ( ( R _FrSe A /\ x e. A ) /\ ch' ) -> E. f ta ) with typecode |-
31 23 30 bnj832 Could not format ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ta ) : No typesetting found for |- ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ta ) with typecode |-
32 bnj645 Could not format ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) -> et ) : No typesetting found for |- ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) -> et ) with typecode |-
33 19.41v ${⊢}\exists {f}\phantom{\rule{.4em}{0ex}}\left({\tau }\wedge {\eta }\right)↔\left(\exists {f}\phantom{\rule{.4em}{0ex}}{\tau }\wedge {\eta }\right)$
34 31 32 33 sylanbrc Could not format ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ( ta /\ et ) ) : No typesetting found for |- ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ( ta /\ et ) ) with typecode |-
35 bnj642 Could not format ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) -> R _FrSe A ) : No typesetting found for |- ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) -> R _FrSe A ) with typecode |-
36 19.41v ${⊢}\exists {f}\phantom{\rule{.4em}{0ex}}\left(\left({\tau }\wedge {\eta }\right)\wedge {R}FrSe{A}\right)↔\left(\exists {f}\phantom{\rule{.4em}{0ex}}\left({\tau }\wedge {\eta }\right)\wedge {R}FrSe{A}\right)$
37 34 35 36 sylanbrc Could not format ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ( ( ta /\ et ) /\ R _FrSe A ) ) : No typesetting found for |- ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ( ( ta /\ et ) /\ R _FrSe A ) ) with typecode |-
38 bnj170 ${⊢}\left({R}FrSe{A}\wedge {\tau }\wedge {\eta }\right)↔\left(\left({\tau }\wedge {\eta }\right)\wedge {R}FrSe{A}\right)$
39 37 38 bnj1198 Could not format ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ( R _FrSe A /\ ta /\ et ) ) : No typesetting found for |- ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ( R _FrSe A /\ ta /\ et ) ) with typecode |-
40 1 6 24 bnj523 Could not format ( ph' <-> ( f  (/) ) = _pred ( x , A , R ) ) : No typesetting found for |- ( ph' <-> ( f  (/) ) = _pred ( x , A , R ) ) with typecode |-
41 2 7 24 bnj539 Could not format ( ps' <-> A. i e. _om ( suc i e. m -> ( f  suc i ) = U_ y e. ( f  i ) _pred ( y , A , R ) ) ) : No typesetting found for |- ( ps' <-> A. i e. _om ( suc i e. m -> ( f  suc i ) = U_ y e. ( f  i ) _pred ( y , A , R ) ) ) with typecode |-
42 40 41 3 12 13 14 bnj544 ${⊢}\left({R}FrSe{A}\wedge {\tau }\wedge {\sigma }\right)\to {G}Fn{n}$
43 14 15 42 bnj561 ${⊢}\left({R}FrSe{A}\wedge {\tau }\wedge {\eta }\right)\to {G}Fn{n}$
44 12 bnj528 ${⊢}{G}\in \mathrm{V}$
45 1 9 44 bnj609 Could not format ( ph" <-> ( G  (/) ) = _pred ( x , A , R ) ) : No typesetting found for |- ( ph" <-> ( G  (/) ) = _pred ( x , A , R ) ) with typecode |-
46 40 3 12 13 14 42 45 bnj545 Could not format ( ( R _FrSe A /\ ta /\ si ) -> ph" ) : No typesetting found for |- ( ( R _FrSe A /\ ta /\ si ) -> ph" ) with typecode |-
47 14 15 46 bnj562 Could not format ( ( R _FrSe A /\ ta /\ et ) -> ph" ) : No typesetting found for |- ( ( R _FrSe A /\ ta /\ et ) -> ph" ) with typecode |-
48 2 10 44 bnj611 Could not format ( ps" <-> A. i e. _om ( suc i e. n -> ( G  suc i ) = U_ y e. ( G  i ) _pred ( y , A , R ) ) ) : No typesetting found for |- ( ps" <-> A. i e. _om ( suc i e. n -> ( G  suc i ) = U_ y e. ( G  i ) _pred ( y , A , R ) ) ) with typecode |-
49 3 12 13 14 15 16 18 19 20 21 22 40 41 42 17 43 48 bnj571 Could not format ( ( R _FrSe A /\ ta /\ et ) -> ps" ) : No typesetting found for |- ( ( R _FrSe A /\ ta /\ et ) -> ps" ) with typecode |-
50 43 47 49 3jca Could not format ( ( R _FrSe A /\ ta /\ et ) -> ( G Fn n /\ ph" /\ ps" ) ) : No typesetting found for |- ( ( R _FrSe A /\ ta /\ et ) -> ( G Fn n /\ ph" /\ ps" ) ) with typecode |-
51 39 50 bnj593 Could not format ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ( G Fn n /\ ph" /\ ps" ) ) : No typesetting found for |- ( ( R _FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ( G Fn n /\ ph" /\ ps" ) ) with typecode |-