ajmoi - Metamath Proof Explorer

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Theorem ajmoi 31347

Description: Every operator has at most one adjoint. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
ip2eqi.1	⊢ 𝑋 = (BaseSet‘𝑈)
ip2eqi.7	⊢ 𝑃 = (·_𝑖OLD‘𝑈)
ip2eqi.u	⊢ 𝑈 ∈ CPreHil_OLD

**Assertion**
Ref	Expression
ajmoi	⊢ ∃*𝑠(𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))

Distinct variable groups: 𝑥,𝑠,𝑃 𝑄,𝑠 𝑥,𝑦,𝑠,𝑇 𝑥,𝑈 𝑋,𝑠,𝑥,𝑦 𝑌,𝑠,𝑥,𝑦 Allowed substitution hints: 𝑃(𝑦) 𝑄(𝑥, 𝑦) 𝑈(𝑦, 𝑠)

Proof of Theorem ajmoi Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.26-2 3149	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)) ∧ ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦))) ↔ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦))))
2		eqtr2 2783	. . . . . . 7 ⊢ ((((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)) ∧ ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦))) → (𝑥𝑃(𝑠‘𝑦)) = (𝑥𝑃(𝑡‘𝑦)))
3	2	2ralimi 3134	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)) ∧ ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦))) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥𝑃(𝑠‘𝑦)) = (𝑥𝑃(𝑡‘𝑦)))
4	1, 3	sylbir 238	. . . . 5 ⊢ ((∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦))) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥𝑃(𝑠‘𝑦)) = (𝑥𝑃(𝑡‘𝑦)))
5		ip2eqi.1	. . . . . . 7 ⊢ 𝑋 = (BaseSet‘𝑈)
6		ip2eqi.7	. . . . . . 7 ⊢ 𝑃 = (·_𝑖OLD‘𝑈)
7		ip2eqi.u	. . . . . . 7 ⊢ 𝑈 ∈ CPreHil_OLD
8	5, 6, 7	phoeqi 31346	. . . . . 6 ⊢ ((𝑠:𝑌⟶𝑋 ∧ 𝑡:𝑌⟶𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥𝑃(𝑠‘𝑦)) = (𝑥𝑃(𝑡‘𝑦)) ↔ 𝑠 = 𝑡))
9	8	biimpa 482	. . . . 5 ⊢ (((𝑠:𝑌⟶𝑋 ∧ 𝑡:𝑌⟶𝑋) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥𝑃(𝑠‘𝑦)) = (𝑥𝑃(𝑡‘𝑦))) → 𝑠 = 𝑡)
10	4, 9	sylan2 605	. . . 4 ⊢ (((𝑠:𝑌⟶𝑋 ∧ 𝑡:𝑌⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦)))) → 𝑠 = 𝑡)
11	10	an4s 673	. . 3 ⊢ (((𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))) ∧ (𝑡:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦)))) → 𝑠 = 𝑡)
12	11	gen2 1829	. 2 ⊢ ∀𝑠∀𝑡(((𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))) ∧ (𝑡:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦)))) → 𝑠 = 𝑡)
13		feq1 6684	. . . 4 ⊢ (𝑠 = 𝑡 → (𝑠:𝑌⟶𝑋 ↔ 𝑡:𝑌⟶𝑋))
14		fveq1 6881	. . . . . . 7 ⊢ (𝑠 = 𝑡 → (𝑠‘𝑦) = (𝑡‘𝑦))
15	14	oveq2d 7433	. . . . . 6 ⊢ (𝑠 = 𝑡 → (𝑥𝑃(𝑠‘𝑦)) = (𝑥𝑃(𝑡‘𝑦)))
16	15	eqeq2d 2773	. . . . 5 ⊢ (𝑠 = 𝑡 → (((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)) ↔ ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦))))
17	16	2ralbidv 3228	. . . 4 ⊢ (𝑠 = 𝑡 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦))))
18	13, 17	anbi12d 644	. . 3 ⊢ (𝑠 = 𝑡 → ((𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))) ↔ (𝑡:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦)))))
19	18	mo4 2593	. 2 ⊢ (∃*𝑠(𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))) ↔ ∀𝑠∀𝑡(((𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))) ∧ (𝑡:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦)))) → 𝑠 = 𝑡))
20	12, 19	mpbir 234	1 ⊢ ∃*𝑠(𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∀wal 1568 = wceq 1570 ∈ wcel 2145 ∃*wmo 2564 ∀wral 3078 ⟶wf 6533 ‘cfv 6537 (class class class)co 7417 BaseSetcba 31075 ·_𝑖OLDcdip 31189 CPreHil_OLDccphlo 31301

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-inf2 9624 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-pre-sup 11206 ax-addf 11207 ax-mulf 11208

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 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 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8163 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-ixp 8909 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-fsupp 9336 df-fi 9385 df-sup 9416 df-inf 9417 df-oi 9486 df-card 9948 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-div 11900 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-7 12336 df-8 12337 df-9 12338 df-n0 12533 df-z 12620 df-dec 12741 df-uz 12892 df-q 13002 df-rp 13047 df-xneg 13167 df-xadd 13168 df-xmul 13169 df-ioo 13406 df-icc 13409 df-fz 13566 df-fzo 13714 df-seq 14070 df-exp 14130 df-hash 14399 df-cj 15190 df-re 15191 df-im 15192 df-sqrt 15326 df-abs 15327 df-clim 15579 df-sum 15778 df-struct 17245 df-sets 17262 df-slot 17280 df-ndx 17292 df-base 17308 df-ress 17329 df-plusg 17361 df-mulr 17362 df-starv 17363 df-sca 17364 df-vsca 17365 df-ip 17366 df-tset 17367 df-ple 17368 df-ds 17370 df-unif 17371 df-hom 17372 df-cco 17373 df-rest 17513 df-topn 17514 df-0g 17532 df-gsum 17533 df-topgen 17534 df-pt 17535 df-prds 17538 df-xrs 17594 df-qtop 17599 df-imas 17600 df-xps 17602 df-mre 17676 df-mrc 17677 df-acs 17679 df-mgm 18736 df-sgrp 18827 df-mnd 18843 df-submnd 18898 df-mulg 19197 df-cntz 19450 df-cmn 19915 df-psmet 21583 df-xmet 21584 df-met 21585 df-bl 21586 df-mopn 21587 df-cnfld 21592 df-top 23125 df-topon 23142 df-topsp 23164 df-bases 23177 df-cld 23250 df-ntr 23251 df-cls 23252 df-cn 23458 df-cnp 23459 df-t1 23545 df-haus 23546 df-tx 23794 df-hmeo 23987 df-xms 24552 df-ms 24553 df-tms 24554 df-grpo 30982 df-gid 30983 df-ginv 30984 df-gdiv 30985 df-ablo 31034 df-vc 31048 df-nv 31081 df-va 31084 df-ba 31085 df-sm 31086 df-0v 31087 df-vs 31088 df-nmcv 31089 df-ims 31090 df-dip 31190 df-ph 31302

This theorem is used by: ajfuni 31348

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