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Theorem prdsbasprj 17491

Description: Each point in a structure product restricts on each coordinate to the relevant base set. (Contributed by Stefan O'Rear, 10-Jan-2015.)

**Hypotheses**
Ref	Expression
prdsbasmpt.y	⊢ 𝑌 = (𝑆X_s𝑅)
prdsbasmpt.b	⊢ 𝐵 = (Base‘𝑌)
prdsbasmpt.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
prdsbasmpt.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
prdsbasmpt.r	⊢ (𝜑 → 𝑅 Fn 𝐼)
prdsbasmpt.t	⊢ (𝜑 → 𝑇 ∈ 𝐵)
prdsbasprj.j	⊢ (𝜑 → 𝐽 ∈ 𝐼)

**Assertion**
Ref	Expression
prdsbasprj	⊢ (𝜑 → (𝑇‘𝐽) ∈ (Base‘(𝑅‘𝐽)))

Proof of Theorem prdsbasprj Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6881	. . 3 ⊢ (𝑥 = 𝐽 → (𝑇‘𝑥) = (𝑇‘𝐽))
2		2fveq3 6886	. . 3 ⊢ (𝑥 = 𝐽 → (Base‘(𝑅‘𝑥)) = (Base‘(𝑅‘𝐽)))
3	1, 2	eleq12d 2829	. 2 ⊢ (𝑥 = 𝐽 → ((𝑇‘𝑥) ∈ (Base‘(𝑅‘𝑥)) ↔ (𝑇‘𝐽) ∈ (Base‘(𝑅‘𝐽))))
4		prdsbasmpt.t	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝐵)
5		prdsbasmpt.y	. . . . 5 ⊢ 𝑌 = (𝑆X_s𝑅)
6		prdsbasmpt.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑌)
7		prdsbasmpt.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
8		prdsbasmpt.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
9		prdsbasmpt.r	. . . . 5 ⊢ (𝜑 → 𝑅 Fn 𝐼)
10	5, 6, 7, 8, 9	prdsbas2 17488	. . . 4 ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
11	4, 10	eleqtrd 2837	. . 3 ⊢ (𝜑 → 𝑇 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
12		elixp2 8920	. . . 4 ⊢ (𝑇 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) ↔ (𝑇 ∈ V ∧ 𝑇 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑇‘𝑥) ∈ (Base‘(𝑅‘𝑥))))
13	12	simp3bi 1147	. . 3 ⊢ (𝑇 ∈ X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) → ∀𝑥 ∈ 𝐼 (𝑇‘𝑥) ∈ (Base‘(𝑅‘𝑥)))
14	11, 13	syl 17	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝑇‘𝑥) ∈ (Base‘(𝑅‘𝑥)))
15		prdsbasprj.j	. 2 ⊢ (𝜑 → 𝐽 ∈ 𝐼)
16	3, 14, 15	rspcdva 3607	1 ⊢ (𝜑 → (𝑇‘𝐽) ∈ (Base‘(𝑅‘𝐽)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ∀wral 3052 Vcvv 3464 Fn wfn 6531 ‘cfv 6536 (class class class)co 7410 Xcixp 8916 Basecbs 17233 X_scprds 17464

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8724 df-map 8847 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9459 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-fz 13530 df-struct 17171 df-slot 17206 df-ndx 17218 df-base 17234 df-plusg 17289 df-mulr 17290 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-hom 17300 df-cco 17301 df-prds 17466

This theorem is referenced by: prdsplusgsgrpcl 18715 prdssgrpd 18716 prdsplusgcl 18751 prdsidlem 18752 prdsmndd 18753 prdspjmhm 18812 prdsinvlem 19037 prdscmnd 19847 prdsmulrngcl 20140 prdsrngd 20141 prdsringd 20286 prdsvscacl 20930 prdslmodd 20931

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