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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 6569 | . . 3 ⊢ (𝐹 = 𝐺 → (Fun 𝐹 ↔ Fun 𝐺)) | |
2 | dmeq 5904 | . . . 4 ⊢ (𝐹 = 𝐺 → dom 𝐹 = dom 𝐺) | |
3 | 2 | eqeq1d 2735 | . . 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 6547 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
6 | df-fn 6547 | . 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 205 ∧ wa 397 = wceq 1542 dom cdm 5677 Fun wfun 6538 Fn wfn 6539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-fun 6546 df-fn 6547 |
This theorem is referenced by: fneq1d 6643 fneq1i 6647 fn0 6682 feq1 6699 foeq1 6802 f1ocnv 6846 dffn5 6951 mpteqb 7018 fnsnb 7164 fnprb 7210 fntpb 7211 eufnfv 7231 frrlem1 8271 frrlem13 8283 wfrlem1OLD 8308 wfrlem3OLDa 8311 wfrlem15OLD 8323 tfrlem12 8389 fsetdmprc0 8849 mapval2 8866 elixp2 8895 ixpfn 8897 elixpsn 8931 inf3lem6 9628 ssttrcl 9710 ttrcltr 9711 ttrclss 9715 ttrclselem2 9721 aceq3lem 10115 dfac4 10117 dfacacn 10136 axcc2lem 10431 axcc3 10433 seqof 14025 ccatvalfn 14531 cshword 14741 0csh0 14743 lmodfopnelem1 20508 rrgsupp 20907 elpt 23076 elptr 23077 ptcmplem3 23558 prdsxmslem2 24038 tgjustr 27725 bnj62 33731 bnj976 33788 bnj66 33871 bnj124 33882 bnj607 33927 bnj873 33935 bnj1234 34024 bnj1463 34066 fineqvac 34097 fnsnbt 41051 eqresfnbd 41054 dssmapf1od 42772 fnchoice 43713 choicefi 43899 axccdom 43921 dfafn5b 45869 rngchomffvalALTV 46893 functhinclem1 47661 |
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