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| Mirrors > Home > MPE Home > Th. List > fneq1i | Structured version Visualization version GIF version | ||
| Description: Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.) |
| Ref | Expression |
|---|---|
| fneq1i.1 | ⊢ 𝐹 = 𝐺 |
| Ref | Expression |
|---|---|
| fneq1i | ⊢ (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fneq1i.1 | . 2 ⊢ 𝐹 = 𝐺 | |
| 2 | fneq1 6659 | . 2 ⊢ (𝐹 = 𝐺 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 Fn wfn 6556 |
| 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-ext 2708 |
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-fun 6563 df-fn 6564 |
| This theorem is referenced by: fnunop 6684 mptfnf 6703 fnopabg 6705 f1oun 6867 f1oi 6886 f1osn 6888 ovid 7574 curry1 8129 curry2 8132 fsplitfpar 8143 frrlem11 8321 wfrlem5OLD 8353 wfrlem13OLD 8361 tfrlem10 8427 tfr1 8437 seqomlem2 8491 seqomlem3 8492 seqomlem4 8493 fnseqom 8495 unblem4 9331 r1fnon 9807 alephfnon 10105 alephfplem4 10147 alephfp 10148 cfsmolem 10310 infpssrlem3 10345 compssiso 10414 hsmexlem5 10470 axdclem2 10560 wunex2 10778 wuncval2 10787 om2uzrani 13993 om2uzf1oi 13994 uzrdglem 13998 uzrdgfni 13999 uzrdg0i 14000 hashkf 14371 dmaf 18094 cdaf 18095 prdsinvlem 19067 srg1zr 20212 pws1 20322 rngcrescrhm 20684 frlmphl 21801 ovolunlem1 25532 0plef 25707 0pledm 25708 itg1ge0 25721 mbfi1fseqlem5 25754 itg2addlem 25793 qaa 26365 precsexlem1 28231 precsexlem2 28232 precsexlem3 28233 precsexlem4 28234 precsexlem5 28235 ex-fpar 30481 0vfval 30625 xrge0pluscn 33939 bnj927 34783 bnj535 34904 fullfunfnv 35947 neibastop2lem 36361 fnmptif 45272 fourierdlem42 46164 fcoreslem4 47078 rngcrescrhmALTV 48196 |
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