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Theorem phoeqi 30838

Description: A condition implying that two operators are equal. (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
phoeqi	⊢ ((𝑆:𝑌⟶𝑋 ∧ 𝑇:𝑌⟶𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥𝑃(𝑆‘𝑦)) = (𝑥𝑃(𝑇‘𝑦)) ↔ 𝑆 = 𝑇))

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

**Proof of Theorem phoeqi**
Step	Hyp	Ref	Expression
1		ralcom 3270	. 2 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥𝑃(𝑆‘𝑦)) = (𝑥𝑃(𝑇‘𝑦)) ↔ ∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 (𝑥𝑃(𝑆‘𝑦)) = (𝑥𝑃(𝑇‘𝑦)))
2		ffvelcdm 7071	. . . . . 6 ⊢ ((𝑆:𝑌⟶𝑋 ∧ 𝑦 ∈ 𝑌) → (𝑆‘𝑦) ∈ 𝑋)
3		ffvelcdm 7071	. . . . . 6 ⊢ ((𝑇:𝑌⟶𝑋 ∧ 𝑦 ∈ 𝑌) → (𝑇‘𝑦) ∈ 𝑋)
4		ip2eqi.1	. . . . . . 7 ⊢ 𝑋 = (BaseSet‘𝑈)
5		ip2eqi.7	. . . . . . 7 ⊢ 𝑃 = (·_𝑖OLD‘𝑈)
6		ip2eqi.u	. . . . . . 7 ⊢ 𝑈 ∈ CPreHil_OLD
7	4, 5, 6	ip2eqi 30837	. . . . . 6 ⊢ (((𝑆‘𝑦) ∈ 𝑋 ∧ (𝑇‘𝑦) ∈ 𝑋) → (∀𝑥 ∈ 𝑋 (𝑥𝑃(𝑆‘𝑦)) = (𝑥𝑃(𝑇‘𝑦)) ↔ (𝑆‘𝑦) = (𝑇‘𝑦)))
8	2, 3, 7	syl2an 596	. . . . 5 ⊢ (((𝑆:𝑌⟶𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑇:𝑌⟶𝑋 ∧ 𝑦 ∈ 𝑌)) → (∀𝑥 ∈ 𝑋 (𝑥𝑃(𝑆‘𝑦)) = (𝑥𝑃(𝑇‘𝑦)) ↔ (𝑆‘𝑦) = (𝑇‘𝑦)))
9	8	anandirs 679	. . . 4 ⊢ (((𝑆:𝑌⟶𝑋 ∧ 𝑇:𝑌⟶𝑋) ∧ 𝑦 ∈ 𝑌) → (∀𝑥 ∈ 𝑋 (𝑥𝑃(𝑆‘𝑦)) = (𝑥𝑃(𝑇‘𝑦)) ↔ (𝑆‘𝑦) = (𝑇‘𝑦)))
10	9	ralbidva 3161	. . 3 ⊢ ((𝑆:𝑌⟶𝑋 ∧ 𝑇:𝑌⟶𝑋) → (∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 (𝑥𝑃(𝑆‘𝑦)) = (𝑥𝑃(𝑇‘𝑦)) ↔ ∀𝑦 ∈ 𝑌 (𝑆‘𝑦) = (𝑇‘𝑦)))
11		ffn 6706	. . . 4 ⊢ (𝑆:𝑌⟶𝑋 → 𝑆 Fn 𝑌)
12		ffn 6706	. . . 4 ⊢ (𝑇:𝑌⟶𝑋 → 𝑇 Fn 𝑌)
13		eqfnfv 7021	. . . 4 ⊢ ((𝑆 Fn 𝑌 ∧ 𝑇 Fn 𝑌) → (𝑆 = 𝑇 ↔ ∀𝑦 ∈ 𝑌 (𝑆‘𝑦) = (𝑇‘𝑦)))
14	11, 12, 13	syl2an 596	. . 3 ⊢ ((𝑆:𝑌⟶𝑋 ∧ 𝑇:𝑌⟶𝑋) → (𝑆 = 𝑇 ↔ ∀𝑦 ∈ 𝑌 (𝑆‘𝑦) = (𝑇‘𝑦)))
15	10, 14	bitr4d 282	. 2 ⊢ ((𝑆:𝑌⟶𝑋 ∧ 𝑇:𝑌⟶𝑋) → (∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 (𝑥𝑃(𝑆‘𝑦)) = (𝑥𝑃(𝑇‘𝑦)) ↔ 𝑆 = 𝑇))
16	1, 15	bitrid 283	1 ⊢ ((𝑆:𝑌⟶𝑋 ∧ 𝑇:𝑌⟶𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥𝑃(𝑆‘𝑦)) = (𝑥𝑃(𝑇‘𝑦)) ↔ 𝑆 = 𝑇))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3051 Fn wfn 6526 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 BaseSetcba 30567 ·_𝑖OLDcdip 30681 CPreHil_OLDccphlo 30793

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207 ax-addf 11208 ax-mulf 11209

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8719 df-map 8842 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-fi 9423 df-sup 9454 df-inf 9455 df-oi 9524 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-uz 12853 df-q 12965 df-rp 13009 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13366 df-icc 13369 df-fz 13525 df-fzo 13672 df-seq 14020 df-exp 14080 df-hash 14349 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-clim 15504 df-sum 15703 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-starv 17286 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-hom 17295 df-cco 17296 df-rest 17436 df-topn 17437 df-0g 17455 df-gsum 17456 df-topgen 17457 df-pt 17458 df-prds 17461 df-xrs 17516 df-qtop 17521 df-imas 17522 df-xps 17524 df-mre 17598 df-mrc 17599 df-acs 17601 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-submnd 18762 df-mulg 19051 df-cntz 19300 df-cmn 19763 df-psmet 21307 df-xmet 21308 df-met 21309 df-bl 21310 df-mopn 21311 df-cnfld 21316 df-top 22832 df-topon 22849 df-topsp 22871 df-bases 22884 df-cld 22957 df-ntr 22958 df-cls 22959 df-cn 23165 df-cnp 23166 df-t1 23252 df-haus 23253 df-tx 23500 df-hmeo 23693 df-xms 24259 df-ms 24260 df-tms 24261 df-grpo 30474 df-gid 30475 df-ginv 30476 df-gdiv 30477 df-ablo 30526 df-vc 30540 df-nv 30573 df-va 30576 df-ba 30577 df-sm 30578 df-0v 30579 df-vs 30580 df-nmcv 30581 df-ims 30582 df-dip 30682 df-ph 30794

This theorem is referenced by: ajmoi 30839

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