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| Mirrors > Home > MPE Home > Th. List > eqfnfvd | Structured version Visualization version GIF version | ||
| Description: Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.) |
| Ref | Expression |
|---|---|
| eqfnfvd.1 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
| eqfnfvd.2 | ⊢ (𝜑 → 𝐺 Fn 𝐴) |
| eqfnfvd.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥)) |
| Ref | Expression |
|---|---|
| eqfnfvd | ⊢ (𝜑 → 𝐹 = 𝐺) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqfnfvd.3 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥)) | |
| 2 | 1 | ralrimiva 3146 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)) |
| 3 | eqfnfvd.1 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
| 4 | eqfnfvd.2 | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴) | |
| 5 | eqfnfv 7051 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) | |
| 6 | 3, 4, 5 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) |
| 7 | 2, 6 | mpbird 257 | 1 ⊢ (𝜑 → 𝐹 = 𝐺) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Fn wfn 6556 ‘cfv 6561 |
| 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 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 |
| This theorem is referenced by: foeqcnvco 7320 f1eqcocnv 7321 offveq 7723 tfrlem1 8416 updjudhcoinlf 9972 updjudhcoinrg 9973 ackbij2lem2 10279 ackbij2lem3 10280 fpwwe2lem7 10677 seqfeq2 14066 seqfeq 14068 seqfeq3 14093 ccatlid 14624 ccatrid 14625 ccatass 14626 ccatswrd 14706 swrdccat2 14707 pfxid 14722 ccatpfx 14739 pfxccat1 14740 swrdswrd 14743 cats1un 14759 swrdccatin1 14763 swrdccatin2 14767 pfxccatin12 14771 revccat 14804 revrev 14805 cshco 14875 swrdco 14876 seqshft 15124 seq1st 16608 xpsfeq 17608 yonedainv 18326 pwsco1mhm 18845 ghmquskerco 19302 f1otrspeq 19465 pmtrfinv 19479 symgtrinv 19490 frgpup3lem 19795 ablfac1eu 20093 zrinitorngc 20642 zrtermorngc 20643 zrtermoringc 20675 psgndiflemB 21618 frlmup1 21818 frlmup3 21820 frlmup4 21821 psrlidm 21982 psrridm 21983 psrass1 21984 subrgascl 22090 evlslem1 22106 psdmplcl 22166 psdvsca 22168 mavmulass 22555 upxp 23631 uptx 23633 cnextfres1 24076 ovolshftlem1 25544 volsup 25591 dvidlem 25950 dvrec 25993 dveq0 26039 dv11cn 26040 ftc1cn 26084 coemulc 26294 aannenlem1 26370 ulmuni 26435 ulmdv 26446 ostthlem1 27671 nvinvfval 30659 sspn 30755 kbass2 32136 xppreima2 32661 fdifsuppconst 32698 indpreima 32850 psgnfzto1stlem 33120 cycpmco2 33153 cyc3co2 33160 ply1gsumz 33619 esumcvg 34087 signstres 34590 hgt750lemb 34671 revpfxsfxrev 35121 subfacp1lem4 35188 cvmliftmolem2 35287 msubff1 35561 iprodefisumlem 35740 poimirlem8 37635 poimirlem13 37640 poimirlem14 37641 ftc1cnnc 37699 eqlkr3 39102 cdleme51finvN 40558 sticksstones11 42157 aks6d1c6lem4 42174 metakunt33 42238 ofun 42277 frlmvscadiccat 42516 fiabv 42546 evlsvvval 42573 fsuppind 42600 ismrcd2 42710 ofoafo 43369 ofoaid1 43371 ofoaid2 43372 ofoaass 43373 ofoacom 43374 naddcnffo 43377 naddcnfcom 43379 naddcnfid1 43380 naddcnfass 43382 rfovcnvf1od 44017 dssmapntrcls 44141 dvconstbi 44353 fsumsermpt 45594 icccncfext 45902 voliooicof 46011 etransclem35 46284 rrxsnicc 46315 ovolval4lem1 46664 fcores 47079 1arymaptf1 48563 2arymaptf1 48574 tposideq 48788 fucoid 49043 |
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