Theorem pmtrdifwrdellem1

Description: Lemma 1 for pmtrdifwrdel . (Contributed by AV, 15-Jan-2019)

	Ref	Expression
Hypotheses	pmtrdifel.t	$$⊢T=\mathrm{ran}\mathrm{pmTrsp}\left(N\setminus \left\{K\right\}\right)$$
	pmtrdifel.r	$$⊢R=\mathrm{ran}\mathrm{pmTrsp}\left(N\right)$$
	pmtrdifwrdel.0	$$⊢U=\left(x\in \left(0..^\left|W\right|\right)⟼\mathrm{pmTrsp}\left(N\right)\left(\mathrm{dom}\left(W\left(x\right)\setminus \mathrm{I}\right)\right)\right)$$
Assertion	pmtrdifwrdellem1	$$⊢W\in \mathrm{Word}T\to U\in \mathrm{Word}R$$

Proof

Step	Hyp	Ref	Expression
1		pmtrdifel.t	$$⊢T=\mathrm{ran}\mathrm{pmTrsp}\left(N\setminus \left\{K\right\}\right)$$
2		pmtrdifel.r	$$⊢R=\mathrm{ran}\mathrm{pmTrsp}\left(N\right)$$
3		pmtrdifwrdel.0	$$⊢U=\left(x\in \left(0..^\left|W\right|\right)⟼\mathrm{pmTrsp}\left(N\right)\left(\mathrm{dom}\left(W\left(x\right)\setminus \mathrm{I}\right)\right)\right)$$
4		wrdsymbcl	$$⊢\left(W\in \mathrm{Word}T\wedge x\in \left(0..^\left|W\right|\right)\right)\to W\left(x\right)\in T$$
5		eqid	$$⊢\mathrm{pmTrsp}\left(N\right)\left(\mathrm{dom}\left(W\left(x\right)\setminus \mathrm{I}\right)\right)=\mathrm{pmTrsp}\left(N\right)\left(\mathrm{dom}\left(W\left(x\right)\setminus \mathrm{I}\right)\right)$$
6	1 2 5	pmtrdifellem1	$$⊢W\left(x\right)\in T\to \mathrm{pmTrsp}\left(N\right)\left(\mathrm{dom}\left(W\left(x\right)\setminus \mathrm{I}\right)\right)\in R$$
7	4 6	syl	$$⊢\left(W\in \mathrm{Word}T\wedge x\in \left(0..^\left|W\right|\right)\right)\to \mathrm{pmTrsp}\left(N\right)\left(\mathrm{dom}\left(W\left(x\right)\setminus \mathrm{I}\right)\right)\in R$$
8	7 3	fmptd	$$⊢W\in \mathrm{Word}T\to U:\left(0..^\left|W\right|\right)⟶R$$
9		iswrdi	$$⊢U:\left(0..^\left|W\right|\right)⟶R\to U\in \mathrm{Word}R$$
10	8 9	syl	$$⊢W\in \mathrm{Word}T\to U\in \mathrm{Word}R$$