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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 5908 | . . 3 ⊢ (𝐹 = 𝐺 → (Fun 𝐹 ↔ Fun 𝐺)) | |
2 | dmeq 5324 | . . . 4 ⊢ (𝐹 = 𝐺 → dom 𝐹 = dom 𝐺) | |
3 | 2 | eqeq1d 2624 | . . 3 ⊢ (𝐹 = 𝐺 → (dom 𝐹 = 𝐴 ↔ dom 𝐺 = 𝐴)) |
4 | 1, 3 | anbi12d 747 | . 2 ⊢ (𝐹 = 𝐺 → ((Fun 𝐹 ∧ dom 𝐹 = 𝐴) ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐴))) |
5 | df-fn 5891 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
6 | df-fn 5891 | . 2 ⊢ (𝐺 Fn 𝐴 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐴)) | |
7 | 4, 5, 6 | 3bitr4g 303 | 1 ⊢ (𝐹 = 𝐺 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 dom cdm 5114 Fun wfun 5882 Fn wfn 5883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-fun 5890 df-fn 5891 |
This theorem is referenced by: fneq1d 5981 fneq1i 5985 fn0 6011 feq1 6026 foeq1 6111 f1ocnv 6149 dffn5 6241 mpteqb 6299 fnsnb 6432 fnprb 6472 fntpb 6473 eufnfv 6491 wfrlem1 7414 wfrlem3a 7417 wfrlem15 7429 tfrlem12 7485 mapval2 7887 elixp2 7912 ixpfn 7914 elixpsn 7947 inf3lem6 8530 aceq3lem 8943 dfac4 8945 dfacacn 8963 axcc2lem 9258 axcc3 9260 seqof 12858 ccatvalfn 13365 cshword 13537 0csh0 13539 lmodfopnelem1 18899 rrgsupp 19291 elpt 21375 elptr 21376 ptcmplem3 21858 prdsxmslem2 22334 bnj62 30786 bnj976 30848 bnj66 30930 bnj124 30941 bnj607 30986 bnj873 30994 bnj1234 31081 bnj1463 31123 frrlem1 31780 dssmapf1od 38315 fnchoice 39188 choicefi 39392 axccdom 39416 dfafn5b 41241 cshword2 41437 rngchomffvalALTV 41995 |
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