Theorem riotasv2s 34244

Description: The value of description binder D for a single-valued class expression C ( y ) (as in e.g. reusv2 4874) in the form of a substitution instance. Special case of riota2f 6632. (Contributed by NM, 3-Mar-2013.) (Proof shortened by Mario Carneiro, 6-Dec-2016.)

Hypothesis

Ref	Expression
riotasv2s.2	|- D = ( iota_ x e. A A. y e. B ( ph -> x = C ) )

Assertion

Ref	Expression
riotasv2s	|- ( ( A e. V /\ D e. A /\ ( E e. B /\ [. E / y ]. ph ) ) -> D = [_ E / y ]_ C )

Distinct variable groups:   x, y, A   x, B, y   x, C   x, E, y   ph, x

Allowed substitution hints:   ph( y)   C( y)   D( x, y)   V( x, y)

Proof of Theorem riotasv2s

Step	Hyp	Ref	Expression
1		3simpc 1060	. 2 |- ( ( A e. V /\ D e. A /\ ( E e. B /\ [. E / y ]. ph ) ) -> ( D e. A /\ ( E e. B /\ [. E / y ]. ph ) ) )
2		simp1 1061	. 2 |- ( ( A e. V /\ D e. A /\ ( E e. B /\ [. E / y ]. ph ) ) -> A e. V )
3		riotasv2s.2	. . . . . 6 |- D = ( iota_ x e. A A. y e. B ( ph -> x = C ) )
4		nfra1 2941	. . . . . . 7 |- F/ y A. y e. B ( ph -> x = C )
5		nfcv 2764	. . . . . . 7 |- F/_ y A
6	4, 5	nfriota 6620	. . . . . 6 |- F/_ y ( iota_ x e. A A. y e. B ( ph -> x = C ) )
7	3, 6	nfcxfr 2762	. . . . 5 |- F/_ y D
8	7	nfel1 2779	. . . 4 |- F/ y D e. A
9		nfv 1843	. . . . 5 |- F/ y E e. B
10		nfsbc1v 3455	. . . . 5 |- F/ y [. E / y ]. ph
11	9, 10	nfan 1828	. . . 4 |- F/ y ( E e. B /\ [. E / y ]. ph )
12	8, 11	nfan 1828	. . 3 |- F/ y ( D e. A /\ ( E e. B /\ [. E / y ]. ph ) )
13		nfcsb1v 3549	. . . 4 |- F/_ y [_ E / y ]_ C
14	13	a1i 11	. . 3 |- ( ( D e. A /\ ( E e. B /\ [. E / y ]. ph ) ) -> F/_ y [_ E / y ]_ C )
15	10	a1i 11	. . 3 |- ( ( D e. A /\ ( E e. B /\ [. E / y ]. ph ) ) -> F/ y [. E / y ]. ph )
16	3	a1i 11	. . 3 |- ( ( D e. A /\ ( E e. B /\ [. E / y ]. ph ) ) -> D = ( iota_ x e. A A. y e. B ( ph -> x = C ) ) )
17		sbceq1a 3446	. . . 4 |- ( y = E -> ( ph <-> [. E / y ]. ph ) )
18	17	adantl 482	. . 3 |- ( ( ( D e. A /\ ( E e. B /\ [. E / y ]. ph ) ) /\ y = E ) -> ( ph <-> [. E / y ]. ph ) )
19		csbeq1a 3542	. . . 4 |- ( y = E -> C = [_ E / y ]_ C )
20	19	adantl 482	. . 3 |- ( ( ( D e. A /\ ( E e. B /\ [. E / y ]. ph ) ) /\ y = E ) -> C = [_ E / y ]_ C )
21		simpl 473	. . 3 |- ( ( D e. A /\ ( E e. B /\ [. E / y ]. ph ) ) -> D e. A )
22		simprl 794	. . 3 |- ( ( D e. A /\ ( E e. B /\ [. E / y ]. ph ) ) -> E e. B )
23		simprr 796	. . 3 |- ( ( D e. A /\ ( E e. B /\ [. E / y ]. ph ) ) -> [. E / y ]. ph )
24	12, 14, 15, 16, 18, 20, 21, 22, 23	riotasv2d 34243	. 2 |- ( ( ( D e. A /\ ( E e. B /\ [. E / y ]. ph ) ) /\ A e. V ) -> D = [_ E / y ]_ C )
25	1, 2, 24	syl2anc 693	1 |- ( ( A e. V /\ D e. A /\ ( E e. B /\ [. E / y ]. ph ) ) -> D = [_ E / y ]_ C )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   A.wral 2912   [.wsbc 3435   [_csb 3533   iota_crio 6610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-undef 7399

This theorem is referenced by: (None)