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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 6437 | . 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 207 = wceq 1528 Fn wfn 6343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-fun 6350 df-fn 6351 |
This theorem is referenced by: fnunsn 6457 mptfnf 6476 fnopabg 6478 f1oun 6627 f1oi 6645 f1osn 6647 ovid 7280 curry1 7788 curry2 7791 fsplitfpar 7803 wfrlem5 7948 wfrlem13 7956 tfrlem10 8012 tfr1 8022 seqomlem2 8076 seqomlem3 8077 seqomlem4 8078 fnseqom 8080 unblem4 8761 r1fnon 9184 alephfnon 9479 alephfplem4 9521 alephfp 9522 cfsmolem 9680 infpssrlem3 9715 compssiso 9784 hsmexlem5 9840 axdclem2 9930 wunex2 10148 wuncval2 10157 om2uzrani 13308 om2uzf1oi 13309 uzrdglem 13313 uzrdgfni 13314 uzrdg0i 13315 hashkf 13680 dmaf 17297 cdaf 17298 prdsinvlem 18146 srg1zr 19208 pws1 19295 frlmphl 20853 ovolunlem1 24025 0plef 24200 0pledm 24201 itg1ge0 24214 itg1addlem4 24227 mbfi1fseqlem5 24247 itg2addlem 24286 qaa 24839 ex-fpar 28168 0vfval 28310 xrge0pluscn 31082 bnj927 31939 bnj535 32061 frrlem11 33030 fullfunfnv 33304 neibastop2lem 33605 fourierdlem42 42311 rngcrescrhm 44284 rngcrescrhmALTV 44302 |
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