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Mirrors > Home > ILE Home > Th. List > fveq1 | GIF version |
Description: Equality theorem for function value. (Contributed by NM, 29-Dec-1996.) |
Ref | Expression |
---|---|
fveq1 | ⊢ (𝐹 = 𝐺 → (𝐹‘𝐴) = (𝐺‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq 3901 | . . 3 ⊢ (𝐹 = 𝐺 → (𝐴𝐹𝑥 ↔ 𝐴𝐺𝑥)) | |
2 | 1 | iotabidv 5079 | . 2 ⊢ (𝐹 = 𝐺 → (℩𝑥𝐴𝐹𝑥) = (℩𝑥𝐴𝐺𝑥)) |
3 | df-fv 5101 | . 2 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥) | |
4 | df-fv 5101 | . 2 ⊢ (𝐺‘𝐴) = (℩𝑥𝐴𝐺𝑥) | |
5 | 2, 3, 4 | 3eqtr4g 2175 | 1 ⊢ (𝐹 = 𝐺 → (𝐹‘𝐴) = (𝐺‘𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 class class class wbr 3899 ℩cio 5056 ‘cfv 5093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rex 2399 df-uni 3707 df-br 3900 df-iota 5058 df-fv 5101 |
This theorem is referenced by: fveq1i 5390 fveq1d 5391 fvmptdf 5476 fvmptdv2 5478 isoeq1 5670 oveq 5748 offval 5957 ofrfval 5958 offval3 6000 smoeq 6155 recseq 6171 tfr0dm 6187 tfrlemiex 6196 tfr1onlemex 6212 tfr1onlemaccex 6213 tfrcllemsucaccv 6219 tfrcllembxssdm 6221 tfrcllemex 6225 tfrcllemaccex 6226 tfrcllemres 6227 rdgeq1 6236 rdgivallem 6246 rdgon 6251 rdg0 6252 frec0g 6262 freccllem 6267 frecfcllem 6269 frecsuclem 6271 frecsuc 6272 mapsncnv 6557 elixp2 6564 elixpsn 6597 mapsnen 6673 mapxpen 6710 ac6sfi 6760 updjud 6935 enomnilem 6978 finomni 6980 exmidomni 6982 fodjuomnilemres 6988 infnninf 6990 nnnninf 6991 ismkvnex 6997 mkvprop 7000 fodjumkvlemres 7001 1fv 9884 seqeq3 10191 iseqf1olemjpcl 10236 iseqf1olemqpcl 10237 iseqf1olemfvp 10238 seq3f1olemqsum 10241 seq3f1olemstep 10242 seq3f1olemp 10243 shftvalg 10576 shftval4g 10577 clim 11018 summodc 11120 fsum3 11124 ennnfonelemim 11864 ctinfom 11868 strnfvnd 11906 iscnp 12295 upxp 12368 elcncf 12656 reldvg 12744 subctctexmid 13123 0nninf 13124 nninff 13125 nnsf 13126 peano4nninf 13127 peano3nninf 13128 nninfalllemn 13129 nninfalllem1 13130 nninfself 13136 nninfsellemeq 13137 nninfsellemeqinf 13139 isomninnlem 13152 trilpolemlt1 13161 |
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