ajmoi - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ajmoi		Structured version Visualization version GIF version

Theorem ajmoi 31115

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 3150	. . . . . 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 2786	. . . . . . 7 ⊢ ((((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)) ∧ ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦))) → (𝑥𝑃(𝑠‘𝑦)) = (𝑥𝑃(𝑡‘𝑦)))
3	2	2ralimi 3135	. . . . . 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 31114	. . . . . 6 ⊢ ((𝑠:𝑌⟶𝑋 ∧ 𝑡:𝑌⟶𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥𝑃(𝑠‘𝑦)) = (𝑥𝑃(𝑡‘𝑦)) ↔ 𝑠 = 𝑡))
9	8	biimpa 481	. . . . 5 ⊢ (((𝑠:𝑌⟶𝑋 ∧ 𝑡:𝑌⟶𝑋) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥𝑃(𝑠‘𝑦)) = (𝑥𝑃(𝑡‘𝑦))) → 𝑠 = 𝑡)
10	4, 9	sylan2 604	. . . 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 672	. . 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 1819	. 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 6673	. . . 4 ⊢ (𝑠 = 𝑡 → (𝑠:𝑌⟶𝑋 ↔ 𝑡:𝑌⟶𝑋))
14		fveq1 6870	. . . . . . 7 ⊢ (𝑠 = 𝑡 → (𝑠‘𝑦) = (𝑡‘𝑦))
15	14	oveq2d 7416	. . . . . 6 ⊢ (𝑠 = 𝑡 → (𝑥𝑃(𝑠‘𝑦)) = (𝑥𝑃(𝑡‘𝑦)))
16	15	eqeq2d 2776	. . . . 5 ⊢ (𝑠 = 𝑡 → (((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)) ↔ ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦))))
17	16	2ralbidv 3229	. . . 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 643	. . 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 2596	. 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

Syntax hints: → wi 4 ∧ wa 400 ∀wal 1561 = wceq 1563 ∈ wcel 2145 ∃*wmo 2567 ∀wral 3079 ⟶wf 6521 ‘cfv 6525 (class class class)co 7400 BaseSetcba 30843 ·_𝑖OLDcdip 30957 CPreHil_OLDccphlo 31069

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5231 ax-sep 5250 ax-nul 5260 ax-pow 5326 ax-pr 5394 ax-un 7722 ax-inf2 9598 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166 ax-addf 11167 ax-mulf 11168

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5105 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6291 df-ord 6352 df-on 6353 df-lim 6354 df-suc 6355 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-om 7851 df-1st 7974 df-2nd 7975 df-supp 8145 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-er 8682 df-map 8814 df-ixp 8884 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12222 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12493 df-z 12580 df-dec 12700 df-uz 12851 df-q 12961 df-rp 13005 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-ioo 13364 df-icc 13367 df-fz 13524 df-fzo 13671 df-seq 14026 df-exp 14086 df-hash 14355 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15527 df-sum 15726 df-struct 17195 df-sets 17212 df-slot 17230 df-ndx 17242 df-base 17258 df-ress 17279 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17463 df-topn 17464 df-0g 17482 df-gsum 17483 df-topgen 17484 df-pt 17485 df-prds 17488 df-xrs 17544 df-qtop 17549 df-imas 17550 df-xps 17552 df-mre 17626 df-mrc 17627 df-acs 17629 df-mgm 18686 df-sgrp 18765 df-mnd 18781 df-submnd 18830 df-mulg 19122 df-cntz 19375 df-cmn 19840 df-psmet 21471 df-xmet 21472 df-met 21473 df-bl 21474 df-mopn 21475 df-cnfld 21480 df-top 23008 df-topon 23025 df-topsp 23047 df-bases 23060 df-cld 23133 df-ntr 23134 df-cls 23135 df-cn 23341 df-cnp 23342 df-t1 23428 df-haus 23429 df-tx 23676 df-hmeo 23869 df-xms 24434 df-ms 24435 df-tms 24436 df-grpo 30750 df-gid 30751 df-ginv 30752 df-gdiv 30753 df-ablo 30802 df-vc 30816 df-nv 30849 df-va 30852 df-ba 30853 df-sm 30854 df-0v 30855 df-vs 30856 df-nmcv 30857 df-ims 30858 df-dip 30958 df-ph 31070

This theorem is referenced by: ajfuni 31116

W3C validator