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Theorem ajfuni 31059

Description: The adjoint function is a function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
ajfuni.5	⊢ 𝐴 = (𝑈adj𝑊)
ajfuni.u	⊢ 𝑈 ∈ CPreHil_OLD
ajfuni.w	⊢ 𝑊 ∈ NrmCVec

**Assertion**
Ref	Expression
ajfuni	⊢ Fun 𝐴

Proof of Theorem ajfuni Dummy variables 𝑡 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funopab 6556	. . 3 ⊢ (Fun {⟨𝑡, 𝑠⟩ ∣ (𝑡:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑠:(BaseSet‘𝑊)⟶(BaseSet‘𝑈) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ (BaseSet‘𝑊)((𝑡‘𝑥)(·_𝑖OLD‘𝑊)𝑦) = (𝑥(·_𝑖OLD‘𝑈)(𝑠‘𝑦)))} ↔ ∀𝑡∃*𝑠(𝑡:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑠:(BaseSet‘𝑊)⟶(BaseSet‘𝑈) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ (BaseSet‘𝑊)((𝑡‘𝑥)(·_𝑖OLD‘𝑊)𝑦) = (𝑥(·_𝑖OLD‘𝑈)(𝑠‘𝑦))))
2		eqid 2762	. . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
3		eqid 2762	. . . . 5 ⊢ (·_𝑖OLD‘𝑈) = (·_𝑖OLD‘𝑈)
4		ajfuni.u	. . . . 5 ⊢ 𝑈 ∈ CPreHil_OLD
5	2, 3, 4	ajmoi 31058	. . . 4 ⊢ ∃*𝑠(𝑠:(BaseSet‘𝑊)⟶(BaseSet‘𝑈) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ (BaseSet‘𝑊)((𝑡‘𝑥)(·_𝑖OLD‘𝑊)𝑦) = (𝑥(·_𝑖OLD‘𝑈)(𝑠‘𝑦)))
6		3simpc 1163	. . . . 5 ⊢ ((𝑡:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑠:(BaseSet‘𝑊)⟶(BaseSet‘𝑈) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ (BaseSet‘𝑊)((𝑡‘𝑥)(·_𝑖OLD‘𝑊)𝑦) = (𝑥(·_𝑖OLD‘𝑈)(𝑠‘𝑦))) → (𝑠:(BaseSet‘𝑊)⟶(BaseSet‘𝑈) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ (BaseSet‘𝑊)((𝑡‘𝑥)(·_𝑖OLD‘𝑊)𝑦) = (𝑥(·_𝑖OLD‘𝑈)(𝑠‘𝑦))))
7	6	moimi 2572	. . . 4 ⊢ (∃𝑠(𝑠:(BaseSet‘𝑊)⟶(BaseSet‘𝑈) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ (BaseSet‘𝑊)((𝑡‘𝑥)(·_𝑖OLD‘𝑊)𝑦) = (𝑥(·_𝑖OLD‘𝑈)(𝑠‘𝑦))) → ∃𝑠(𝑡:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑠:(BaseSet‘𝑊)⟶(BaseSet‘𝑈) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ (BaseSet‘𝑊)((𝑡‘𝑥)(·_𝑖OLD‘𝑊)𝑦) = (𝑥(·_𝑖OLD‘𝑈)(𝑠‘𝑦))))
8	5, 7	ax-mp 5	. . 3 ⊢ ∃*𝑠(𝑡:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑠:(BaseSet‘𝑊)⟶(BaseSet‘𝑈) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ (BaseSet‘𝑊)((𝑡‘𝑥)(·_𝑖OLD‘𝑊)𝑦) = (𝑥(·_𝑖OLD‘𝑈)(𝑠‘𝑦)))
9	1, 8	mpgbir 1819	. 2 ⊢ Fun {⟨𝑡, 𝑠⟩ ∣ (𝑡:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑠:(BaseSet‘𝑊)⟶(BaseSet‘𝑈) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ (BaseSet‘𝑊)((𝑡‘𝑥)(·_𝑖OLD‘𝑊)𝑦) = (𝑥(·_𝑖OLD‘𝑈)(𝑠‘𝑦)))}
10	4	phnvi 31016	. . . 4 ⊢ 𝑈 ∈ NrmCVec
11		ajfuni.w	. . . 4 ⊢ 𝑊 ∈ NrmCVec
12		eqid 2762	. . . . 5 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
13		eqid 2762	. . . . 5 ⊢ (·_𝑖OLD‘𝑊) = (·_𝑖OLD‘𝑊)
14		ajfuni.5	. . . . 5 ⊢ 𝐴 = (𝑈adj𝑊)
15	2, 12, 3, 13, 14	ajfval 31009	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐴 = {⟨𝑡, 𝑠⟩ ∣ (𝑡:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑠:(BaseSet‘𝑊)⟶(BaseSet‘𝑈) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ (BaseSet‘𝑊)((𝑡‘𝑥)(·_𝑖OLD‘𝑊)𝑦) = (𝑥(·_𝑖OLD‘𝑈)(𝑠‘𝑦)))})
16	10, 11, 15	mp2an 702	. . 3 ⊢ 𝐴 = {⟨𝑡, 𝑠⟩ ∣ (𝑡:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑠:(BaseSet‘𝑊)⟶(BaseSet‘𝑈) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ (BaseSet‘𝑊)((𝑡‘𝑥)(·_𝑖OLD‘𝑊)𝑦) = (𝑥(·_𝑖OLD‘𝑈)(𝑠‘𝑦)))}
17	16	funeqi 6542	. 2 ⊢ (Fun 𝐴 ↔ Fun {⟨𝑡, 𝑠⟩ ∣ (𝑡:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑠:(BaseSet‘𝑊)⟶(BaseSet‘𝑈) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ (BaseSet‘𝑊)((𝑡‘𝑥)(·_𝑖OLD‘𝑊)𝑦) = (𝑥(·_𝑖OLD‘𝑈)(𝑠‘𝑦)))})
18	9, 17	mpbir 233	1 ⊢ Fun 𝐴

Colors of variables: wff setvar class

Syntax hints: ∧ wa 399 ∧ w3a 1098 = wceq 1560 ∈ wcel 2142 ∃*wmo 2564 ∀wral 3076 {copab 5162 Fun wfun 6515 ⟶wf 6517 ‘cfv 6521 (class class class)co 7396 NrmCVeccnv 30784 BaseSetcba 30786 ·_𝑖OLDcdip 30900 adjcaj 30948 CPreHil_OLDccphlo 31012

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-inf2 9596 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-pre-sup 11151 ax-addf 11152 ax-mulf 11153

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-iin 4952 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7660 df-om 7847 df-1st 7970 df-2nd 7971 df-supp 8141 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8678 df-map 8810 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9308 df-fi 9357 df-sup 9388 df-inf 9389 df-oi 9458 df-card 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-dec 12689 df-uz 12840 df-q 12950 df-rp 12994 df-xneg 13114 df-xadd 13115 df-xmul 13116 df-ioo 13353 df-icc 13356 df-fz 13513 df-fzo 13660 df-seq 14015 df-exp 14075 df-hash 14344 df-cj 15126 df-re 15127 df-im 15128 df-sqrt 15262 df-abs 15263 df-clim 15515 df-sum 15714 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-mulr 17300 df-starv 17301 df-sca 17302 df-vsca 17303 df-ip 17304 df-tset 17305 df-ple 17306 df-ds 17308 df-unif 17309 df-hom 17310 df-cco 17311 df-rest 17451 df-topn 17452 df-0g 17470 df-gsum 17471 df-topgen 17472 df-pt 17473 df-prds 17476 df-xrs 17532 df-qtop 17537 df-imas 17538 df-xps 17540 df-mre 17614 df-mrc 17615 df-acs 17617 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-submnd 18818 df-mulg 19110 df-cntz 19357 df-cmn 19822 df-psmet 21413 df-xmet 21414 df-met 21415 df-bl 21416 df-mopn 21417 df-cnfld 21422 df-top 22951 df-topon 22968 df-topsp 22990 df-bases 23003 df-cld 23076 df-ntr 23077 df-cls 23078 df-cn 23284 df-cnp 23285 df-t1 23371 df-haus 23372 df-tx 23619 df-hmeo 23812 df-xms 24377 df-ms 24378 df-tms 24379 df-grpo 30693 df-gid 30694 df-ginv 30695 df-gdiv 30696 df-ablo 30745 df-vc 30759 df-nv 30792 df-va 30795 df-ba 30796 df-sm 30797 df-0v 30798 df-vs 30799 df-nmcv 30800 df-ims 30801 df-dip 30901 df-aj 30950 df-ph 31013

This theorem is referenced by: ajfun 31060

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