dprdf2 - Metamath Proof Explorer

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Theorem dprdf2 19964

Description: The function 𝑆 is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)

**Hypotheses**
Ref	Expression
dprdcntz.1	⊢ (𝜑 → 𝐺dom DProd 𝑆)
dprdcntz.2	⊢ (𝜑 → dom 𝑆 = 𝐼)

**Assertion**
Ref	Expression
dprdf2	⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))

**Proof of Theorem dprdf2**
Step	Hyp	Ref	Expression
1		dprdcntz.1	. . 3 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
2		dprdf 19963	. . 3 ⊢ (𝐺dom DProd 𝑆 → 𝑆:dom 𝑆⟶(SubGrp‘𝐺))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑆:dom 𝑆⟶(SubGrp‘𝐺))
4		dprdcntz.2	. . 3 ⊢ (𝜑 → dom 𝑆 = 𝐼)
5	4	feq2d 6708	. 2 ⊢ (𝜑 → (𝑆:dom 𝑆⟶(SubGrp‘𝐺) ↔ 𝑆:𝐼⟶(SubGrp‘𝐺)))
6	3, 5	mpbid 231	1 ⊢ (𝜑 → 𝑆:𝐼⟶(SubGrp‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1534 class class class wbr 5148 dom cdm 5678 ⟶wf 6544 ‘cfv 6548 SubGrpcsubg 19075 DProd cdprd 19950

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-ixp 8917 df-dprd 19952

This theorem is referenced by: dprdff 19969 dprdfid 19974 dprdfinv 19976 dprdfadd 19977 dprdfeq0 19979 dprdres 19985 dprdss 19986 dprdf1o 19989 dprdf1 19990 subgdprd 19992 dmdprdsplitlem 19994 dprdcntz2 19995 dpjlem 20008 dpjcntz 20009 dpjdisj 20010 dpjlsm 20011 dpjf 20014 dpjidcl 20015 dpjlid 20018 dpjghm 20020 dpjghm2 20021 ablfac1c 20028 ablfac1eulem 20029 ablfac1eu 20030 ablfaclem2 20043 ablfaclem3 20044 dchrptlem3 27212

	
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