Metamath Proof Explorer

Theorem bnj938

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 bnj938.1 ${⊢}{D}=\mathrm{\omega }\setminus \left\{\varnothing \right\}$
bnj938.2 No typesetting found for |- ( ta <-> ( f Fn m /\ ph' /\ ps' ) ) with typecode |-
bnj938.3 ${⊢}{\sigma }↔\left({m}\in {D}\wedge {n}=\mathrm{suc}{m}\wedge {p}\in {m}\right)$
bnj938.4 No typesetting found for |- ( ph' <-> ( f  (/) ) = _pred ( X , A , R ) ) with typecode |-
bnj938.5 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 |-
Assertion bnj938 ${⊢}\left({R}FrSe{A}\wedge {X}\in {A}\wedge {\tau }\wedge {\sigma }\right)\to \bigcup _{{y}\in {f}\left({p}\right)}pred\left({y},{A},{R}\right)\in \mathrm{V}$

Proof

Step Hyp Ref Expression
1 bnj938.1 ${⊢}{D}=\mathrm{\omega }\setminus \left\{\varnothing \right\}$
2 bnj938.2 Could not format ( ta <-> ( f Fn m /\ ph' /\ ps' ) ) : No typesetting found for |- ( ta <-> ( f Fn m /\ ph' /\ ps' ) ) with typecode |-
3 bnj938.3 ${⊢}{\sigma }↔\left({m}\in {D}\wedge {n}=\mathrm{suc}{m}\wedge {p}\in {m}\right)$
4 bnj938.4 Could not format ( ph' <-> ( f  (/) ) = _pred ( X , A , R ) ) : No typesetting found for |- ( ph' <-> ( f  (/) ) = _pred ( X , A , R ) ) with typecode |-
5 bnj938.5 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 |-
6 elisset ${⊢}{X}\in {A}\to \exists {x}\phantom{\rule{.4em}{0ex}}{x}={X}$
7 6 bnj706 ${⊢}\left({R}FrSe{A}\wedge {X}\in {A}\wedge {\tau }\wedge {\sigma }\right)\to \exists {x}\phantom{\rule{.4em}{0ex}}{x}={X}$
8 bnj291 ${⊢}\left({R}FrSe{A}\wedge {X}\in {A}\wedge {\tau }\wedge {\sigma }\right)↔\left(\left({R}FrSe{A}\wedge {\tau }\wedge {\sigma }\right)\wedge {X}\in {A}\right)$
9 8 simplbi ${⊢}\left({R}FrSe{A}\wedge {X}\in {A}\wedge {\tau }\wedge {\sigma }\right)\to \left({R}FrSe{A}\wedge {\tau }\wedge {\sigma }\right)$
10 bnj602 ${⊢}{x}={X}\to pred\left({x},{A},{R}\right)=pred\left({X},{A},{R}\right)$
11 10 eqeq2d ${⊢}{x}={X}\to \left({f}\left(\varnothing \right)=pred\left({x},{A},{R}\right)↔{f}\left(\varnothing \right)=pred\left({X},{A},{R}\right)\right)$
12 11 4 syl6bbr Could not format ( x = X -> ( ( f  (/) ) = _pred ( x , A , R ) <-> ph' ) ) : No typesetting found for |- ( x = X -> ( ( f  (/) ) = _pred ( x , A , R ) <-> ph' ) ) with typecode |-
13 12 3anbi2d Could not format ( x = X -> ( ( f Fn m /\ ( f  (/) ) = _pred ( x , A , R ) /\ ps' ) <-> ( f Fn m /\ ph' /\ ps' ) ) ) : No typesetting found for |- ( x = X -> ( ( f Fn m /\ ( f  (/) ) = _pred ( x , A , R ) /\ ps' ) <-> ( f Fn m /\ ph' /\ ps' ) ) ) with typecode |-
14 13 2 syl6bbr Could not format ( x = X -> ( ( f Fn m /\ ( f  (/) ) = _pred ( x , A , R ) /\ ps' ) <-> ta ) ) : No typesetting found for |- ( x = X -> ( ( f Fn m /\ ( f  (/) ) = _pred ( x , A , R ) /\ ps' ) <-> ta ) ) with typecode |-
15 14 3anbi2d Could not format ( x = X -> ( ( R _FrSe A /\ ( f Fn m /\ ( f  (/) ) = _pred ( x , A , R ) /\ ps' ) /\ si ) <-> ( R _FrSe A /\ ta /\ si ) ) ) : No typesetting found for |- ( x = X -> ( ( R _FrSe A /\ ( f Fn m /\ ( f  (/) ) = _pred ( x , A , R ) /\ ps' ) /\ si ) <-> ( R _FrSe A /\ ta /\ si ) ) ) with typecode |-
16 9 15 syl5ibr Could not format ( x = X -> ( ( R _FrSe A /\ X e. A /\ ta /\ si ) -> ( R _FrSe A /\ ( f Fn m /\ ( f  (/) ) = _pred ( x , A , R ) /\ ps' ) /\ si ) ) ) : No typesetting found for |- ( x = X -> ( ( R _FrSe A /\ X e. A /\ ta /\ si ) -> ( R _FrSe A /\ ( f Fn m /\ ( f  (/) ) = _pred ( x , A , R ) /\ ps' ) /\ si ) ) ) with typecode |-
17 biid Could not format ( ( f Fn m /\ ( f  (/) ) = _pred ( x , A , R ) /\ ps' ) <-> ( f Fn m /\ ( f  (/) ) = _pred ( x , A , R ) /\ ps' ) ) : No typesetting found for |- ( ( f Fn m /\ ( f  (/) ) = _pred ( x , A , R ) /\ ps' ) <-> ( f Fn m /\ ( f  (/) ) = _pred ( x , A , R ) /\ ps' ) ) with typecode |-
18 biid ${⊢}{f}\left(\varnothing \right)=pred\left({x},{A},{R}\right)↔{f}\left(\varnothing \right)=pred\left({x},{A},{R}\right)$
19 1 17 3 18 5 bnj546 Could not format ( ( R _FrSe A /\ ( f Fn m /\ ( f  (/) ) = _pred ( x , A , R ) /\ ps' ) /\ si ) -> U_ y e. ( f  p ) _pred ( y , A , R ) e. _V ) : No typesetting found for |- ( ( R _FrSe A /\ ( f Fn m /\ ( f  (/) ) = _pred ( x , A , R ) /\ ps' ) /\ si ) -> U_ y e. ( f ` p ) _pred ( y , A , R ) e. _V ) with typecode |-
20 16 19 syl6 ${⊢}{x}={X}\to \left(\left({R}FrSe{A}\wedge {X}\in {A}\wedge {\tau }\wedge {\sigma }\right)\to \bigcup _{{y}\in {f}\left({p}\right)}pred\left({y},{A},{R}\right)\in \mathrm{V}\right)$
21 20 exlimiv ${⊢}\exists {x}\phantom{\rule{.4em}{0ex}}{x}={X}\to \left(\left({R}FrSe{A}\wedge {X}\in {A}\wedge {\tau }\wedge {\sigma }\right)\to \bigcup _{{y}\in {f}\left({p}\right)}pred\left({y},{A},{R}\right)\in \mathrm{V}\right)$
22 7 21 mpcom ${⊢}\left({R}FrSe{A}\wedge {X}\in {A}\wedge {\tau }\wedge {\sigma }\right)\to \bigcup _{{y}\in {f}\left({p}\right)}pred\left({y},{A},{R}\right)\in \mathrm{V}$