Theorem bnj540

Description: Technical lemma for bnj852 . 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	bnj540.1	$$⊢\psi \leftrightarrow \forall i\in \mathrm{\omega }\left(\mathrm{suc}i\in N\to f\left(\mathrm{suc}i\right)=\bigcup _{y\in f\left(i\right)}pred\left(y,A,R\right)\right)$$
	bnj540.2	No typesetting found for |- ( ps" <-> [. G / f ]. ps ) with typecode |-
	bnj540.3	$$⊢G\in \mathrm{V}$$
Assertion	bnj540	Could not format assertion : 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 |-

Proof

Step	Hyp	Ref	Expression
1		bnj540.1	$$⊢\psi \leftrightarrow \forall i\in \mathrm{\omega }\left(\mathrm{suc}i\in N\to f\left(\mathrm{suc}i\right)=\bigcup _{y\in f\left(i\right)}pred\left(y,A,R\right)\right)$$
2		bnj540.2	Could not format ( ps" <-> [. G / f ]. ps ) : No typesetting found for |- ( ps" <-> [. G / f ]. ps ) with typecode |-
3		bnj540.3	$$⊢G\in \mathrm{V}$$
4	1	sbcbii	$$⊢\underset{\dot{}}{[}G/f\underset{\dot{}}{]}\psi \leftrightarrow \underset{\dot{}}{[}G/f\underset{\dot{}}{]}\forall i\in \mathrm{\omega }\left(\mathrm{suc}i\in N\to f\left(\mathrm{suc}i\right)=\bigcup _{y\in f\left(i\right)}pred\left(y,A,R\right)\right)$$
5	3	bnj538	$$⊢\underset{\dot{}}{[}G/f\underset{\dot{}}{]}\forall i\in \mathrm{\omega }\left(\mathrm{suc}i\in N\to f\left(\mathrm{suc}i\right)=\bigcup _{y\in f\left(i\right)}pred\left(y,A,R\right)\right)\leftrightarrow \forall i\in \mathrm{\omega }\underset{\dot{}}{[}G/f\underset{\dot{}}{]}\left(\mathrm{suc}i\in N\to f\left(\mathrm{suc}i\right)=\bigcup _{y\in f\left(i\right)}pred\left(y,A,R\right)\right)$$
6		sbcimg	$$⊢G\in \mathrm{V}\to \left(\underset{\dot{}}{[}G/f\underset{\dot{}}{]}\left(\mathrm{suc}i\in N\to f\left(\mathrm{suc}i\right)=\bigcup _{y\in f\left(i\right)}pred\left(y,A,R\right)\right)\leftrightarrow \left(\underset{\dot{}}{[}G/f\underset{\dot{}}{]}\mathrm{suc}i\in N\to \underset{\dot{}}{[}G/f\underset{\dot{}}{]}f\left(\mathrm{suc}i\right)=\bigcup _{y\in f\left(i\right)}pred\left(y,A,R\right)\right)\right)$$
7	3 6	ax-mp	$$⊢\underset{\dot{}}{[}G/f\underset{\dot{}}{]}\left(\mathrm{suc}i\in N\to f\left(\mathrm{suc}i\right)=\bigcup _{y\in f\left(i\right)}pred\left(y,A,R\right)\right)\leftrightarrow \left(\underset{\dot{}}{[}G/f\underset{\dot{}}{]}\mathrm{suc}i\in N\to \underset{\dot{}}{[}G/f\underset{\dot{}}{]}f\left(\mathrm{suc}i\right)=\bigcup _{y\in f\left(i\right)}pred\left(y,A,R\right)\right)$$
8	7	ralbii	$$⊢\forall i\in \mathrm{\omega }\underset{\dot{}}{[}G/f\underset{\dot{}}{]}\left(\mathrm{suc}i\in N\to f\left(\mathrm{suc}i\right)=\bigcup _{y\in f\left(i\right)}pred\left(y,A,R\right)\right)\leftrightarrow \forall i\in \mathrm{\omega }\left(\underset{\dot{}}{[}G/f\underset{\dot{}}{]}\mathrm{suc}i\in N\to \underset{\dot{}}{[}G/f\underset{\dot{}}{]}f\left(\mathrm{suc}i\right)=\bigcup _{y\in f\left(i\right)}pred\left(y,A,R\right)\right)$$
9	4 5 8	3bitri	$$⊢\underset{\dot{}}{[}G/f\underset{\dot{}}{]}\psi \leftrightarrow \forall i\in \mathrm{\omega }\left(\underset{\dot{}}{[}G/f\underset{\dot{}}{]}\mathrm{suc}i\in N\to \underset{\dot{}}{[}G/f\underset{\dot{}}{]}f\left(\mathrm{suc}i\right)=\bigcup _{y\in f\left(i\right)}pred\left(y,A,R\right)\right)$$
10	3	bnj525	$$⊢\underset{\dot{}}{[}G/f\underset{\dot{}}{]}\mathrm{suc}i\in N\leftrightarrow \mathrm{suc}i\in N$$
11		fveq1	$$⊢f=G\to f\left(\mathrm{suc}i\right)=G\left(\mathrm{suc}i\right)$$
12		fveq1	$$⊢f=G\to f\left(i\right)=G\left(i\right)$$
13	12	bnj1113	$$⊢f=G\to \bigcup _{y\in f\left(i\right)}pred\left(y,A,R\right)=\bigcup _{y\in G\left(i\right)}pred\left(y,A,R\right)$$
14	11 13	eqeq12d	$$⊢f=G\to \left(f\left(\mathrm{suc}i\right)=\bigcup _{y\in f\left(i\right)}pred\left(y,A,R\right)\leftrightarrow G\left(\mathrm{suc}i\right)=\bigcup _{y\in G\left(i\right)}pred\left(y,A,R\right)\right)$$
15	3 14	sbcie	$$⊢\underset{\dot{}}{[}G/f\underset{\dot{}}{]}f\left(\mathrm{suc}i\right)=\bigcup _{y\in f\left(i\right)}pred\left(y,A,R\right)\leftrightarrow G\left(\mathrm{suc}i\right)=\bigcup _{y\in G\left(i\right)}pred\left(y,A,R\right)$$
16	10 15	imbi12i	$$⊢\left(\underset{\dot{}}{[}G/f\underset{\dot{}}{]}\mathrm{suc}i\in N\to \underset{\dot{}}{[}G/f\underset{\dot{}}{]}f\left(\mathrm{suc}i\right)=\bigcup _{y\in f\left(i\right)}pred\left(y,A,R\right)\right)\leftrightarrow \left(\mathrm{suc}i\in N\to G\left(\mathrm{suc}i\right)=\bigcup _{y\in G\left(i\right)}pred\left(y,A,R\right)\right)$$
17	16	ralbii	$$⊢\forall i\in \mathrm{\omega }\left(\underset{\dot{}}{[}G/f\underset{\dot{}}{]}\mathrm{suc}i\in N\to \underset{\dot{}}{[}G/f\underset{\dot{}}{]}f\left(\mathrm{suc}i\right)=\bigcup _{y\in f\left(i\right)}pred\left(y,A,R\right)\right)\leftrightarrow \forall i\in \mathrm{\omega }\left(\mathrm{suc}i\in N\to G\left(\mathrm{suc}i\right)=\bigcup _{y\in G\left(i\right)}pred\left(y,A,R\right)\right)$$
18	2 9 17	3bitri	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 |-