Theorem riotasv2d 34243

Description: Value of description binder D for a single-valued class expression C ( y ) (as in e.g. reusv2 4874). Special case of riota2f 6632. (Contributed by NM, 2-Mar-2013.)

Hypotheses

Ref	Expression
riotasv2d.1	|- F/ y ph
riotasv2d.2	|- ( ph -> F/_ y F )
riotasv2d.3	|- ( ph -> F/ y ch )
riotasv2d.4	|- ( ph -> D = ( iota_ x e. A A. y e. B ( ps -> x = C ) ) )
riotasv2d.5	|- ( ( ph /\ y = E ) -> ( ps <-> ch ) )
riotasv2d.6	|- ( ( ph /\ y = E ) -> C = F )
riotasv2d.7	|- ( ph -> D e. A )
riotasv2d.8	|- ( ph -> E e. B )
riotasv2d.9	|- ( ph -> ch )

Assertion

Ref	Expression
riotasv2d	|- ( ( ph /\ A e. V ) -> D = F )

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

Allowed substitution hints:   ph( x, y)   ps( y)   ch( x, y)   C( y)   D( x, y)   E( x)   F( x, y)   V( x, y)

Proof of Theorem riotasv2d

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( A e. V -> A e. _V )
2		riotasv2d.8	. . . 4 |- ( ph -> E e. B )
3	2	adantr 481	. . 3 |- ( ( ph /\ A e. _V ) -> E e. B )
4		riotasv2d.9	. . . 4 |- ( ph -> ch )
5	4	adantr 481	. . 3 |- ( ( ph /\ A e. _V ) -> ch )
6		eleq1 2689	. . . . . . . 8 |- ( y = E -> ( y e. B <-> E e. B ) )
7	6	adantl 482	. . . . . . 7 |- ( ( ph /\ y = E ) -> ( y e. B <-> E e. B ) )
8		riotasv2d.5	. . . . . . 7 |- ( ( ph /\ y = E ) -> ( ps <-> ch ) )
9	7, 8	anbi12d 747	. . . . . 6 |- ( ( ph /\ y = E ) -> ( ( y e. B /\ ps ) <-> ( E e. B /\ ch ) ) )
10		riotasv2d.6	. . . . . . 7 |- ( ( ph /\ y = E ) -> C = F )
11	10	eqeq2d 2632	. . . . . 6 |- ( ( ph /\ y = E ) -> ( D = C <-> D = F ) )
12	9, 11	imbi12d 334	. . . . 5 |- ( ( ph /\ y = E ) -> ( ( ( y e. B /\ ps ) -> D = C ) <-> ( ( E e. B /\ ch ) -> D = F ) ) )
13	12	adantlr 751	. . . 4 |- ( ( ( ph /\ A e. _V ) /\ y = E ) -> ( ( ( y e. B /\ ps ) -> D = C ) <-> ( ( E e. B /\ ch ) -> D = F ) ) )
14		riotasv2d.4	. . . . 5 |- ( ph -> D = ( iota_ x e. A A. y e. B ( ps -> x = C ) ) )
15		riotasv2d.7	. . . . 5 |- ( ph -> D e. A )
16	14, 15	riotasvd 34242	. . . 4 |- ( ( ph /\ A e. _V ) -> ( ( y e. B /\ ps ) -> D = C ) )
17		riotasv2d.1	. . . . 5 |- F/ y ph
18		nfv 1843	. . . . 5 |- F/ y A e. _V
19	17, 18	nfan 1828	. . . 4 |- F/ y ( ph /\ A e. _V )
20		nfcvd 2765	. . . 4 |- ( ( ph /\ A e. _V ) -> F/_ y E )
21		nfvd 1844	. . . . . . 7 |- ( ph -> F/ y E e. B )
22		riotasv2d.3	. . . . . . 7 |- ( ph -> F/ y ch )
23	21, 22	nfand 1826	. . . . . 6 |- ( ph -> F/ y ( E e. B /\ ch ) )
24		nfra1 2941	. . . . . . . . 9 |- F/ y A. y e. B ( ps -> x = C )
25		nfcv 2764	. . . . . . . . 9 |- F/_ y A
26	24, 25	nfriota 6620	. . . . . . . 8 |- F/_ y ( iota_ x e. A A. y e. B ( ps -> x = C ) )
27	17, 14	nfceqdf 2760	. . . . . . . 8 |- ( ph -> ( F/_ y D <-> F/_ y ( iota_ x e. A A. y e. B ( ps -> x = C ) ) ) )
28	26, 27	mpbiri 248	. . . . . . 7 |- ( ph -> F/_ y D )
29		riotasv2d.2	. . . . . . 7 |- ( ph -> F/_ y F )
30	28, 29	nfeqd 2772	. . . . . 6 |- ( ph -> F/ y D = F )
31	23, 30	nfimd 1823	. . . . 5 |- ( ph -> F/ y ( ( E e. B /\ ch ) -> D = F ) )
32	31	adantr 481	. . . 4 |- ( ( ph /\ A e. _V ) -> F/ y ( ( E e. B /\ ch ) -> D = F ) )
33	3, 13, 16, 19, 20, 32	vtocldf 3256	. . 3 |- ( ( ph /\ A e. _V ) -> ( ( E e. B /\ ch ) -> D = F ) )
34	3, 5, 33	mp2and 715	. 2 |- ( ( ph /\ A e. _V ) -> D = F )
35	1, 34	sylan2 491	1 |- ( ( ph /\ A e. V ) -> D = F )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   F/_wnfc 2751   A.wral 2912   _Vcvv 3200   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-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:  riotasv2s  34244  cdleme42b  35766