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Mirrors > Home > MPE Home > Th. List > fneq1 | Structured version Visualization version GIF version |
Description: Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.) |
Ref | Expression |
---|---|
fneq1 | ⊢ (𝐹 = 𝐺 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funeq 6588 | . . 3 ⊢ (𝐹 = 𝐺 → (Fun 𝐹 ↔ Fun 𝐺)) | |
2 | dmeq 5917 | . . . 4 ⊢ (𝐹 = 𝐺 → dom 𝐹 = dom 𝐺) | |
3 | 2 | eqeq1d 2737 | . . 3 ⊢ (𝐹 = 𝐺 → (dom 𝐹 = 𝐴 ↔ dom 𝐺 = 𝐴)) |
4 | 1, 3 | anbi12d 632 | . 2 ⊢ (𝐹 = 𝐺 → ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐴))) |
5 | df-fn 6566 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
6 | df-fn 6566 | . 2 ⊢ (𝐺 Fn 𝐴 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐴)) | |
7 | 4, 5, 6 | 3bitr4g 314 | 1 ⊢ (𝐹 = 𝐺 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 dom cdm 5689 Fun wfun 6557 Fn wfn 6558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-fun 6565 df-fn 6566 |
This theorem is referenced by: fneq1d 6662 fneq1i 6666 fn0 6700 feq1 6717 foeq1 6817 f1ocnv 6861 dffn5 6967 mpteqb 7035 fnsnb 7186 fnprb 7228 fntpb 7229 eufnfv 7249 frrlem1 8310 frrlem13 8322 wfrlem1OLD 8347 wfrlem3OLDa 8350 wfrlem15OLD 8362 tfrlem12 8428 fsetdmprc0 8894 mapval2 8911 elixp2 8940 ixpfn 8942 elixpsn 8976 inf3lem6 9671 ssttrcl 9753 ttrcltr 9754 ttrclss 9758 ttrclselem2 9764 aceq3lem 10158 dfac4 10160 dfacacn 10180 axcc2lem 10474 axcc3 10476 seqof 14097 ccatvalfn 14616 cshword 14826 0csh0 14828 rrgsupp 20718 lmodfopnelem1 20913 elpt 23596 elptr 23597 ptcmplem3 24078 prdsxmslem2 24558 tgjustr 28497 bnj62 34713 bnj976 34770 bnj66 34853 bnj124 34864 bnj607 34909 bnj873 34917 bnj1234 35006 bnj1463 35048 fineqvac 35090 gblacfnacd 35092 fnsnbt 42250 eqresfnbd 42252 dssmapf1od 44011 fnchoice 44967 choicefi 45143 axccdom 45165 dfafn5b 47111 rngchomffvalALTV 48122 functhinclem1 48841 |
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