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Mirrors > Home > MPE Home > Th. List > rexrn | Structured version Visualization version GIF version |
Description: Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.) |
Ref | Expression |
---|---|
rexrn.1 | ⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
rexrn | ⊢ (𝐹 Fn 𝐴 → (∃𝑥 ∈ ran 𝐹𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvexd 6660 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ V) | |
2 | fvelrnb 6701 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝑥)) | |
3 | eqcom 2805 | . . . 4 ⊢ ((𝐹‘𝑦) = 𝑥 ↔ 𝑥 = (𝐹‘𝑦)) | |
4 | 3 | rexbii 3210 | . . 3 ⊢ (∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑥 = (𝐹‘𝑦)) |
5 | 2, 4 | syl6bb 290 | . 2 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐴 𝑥 = (𝐹‘𝑦))) |
6 | rexrn.1 | . . 3 ⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓)) | |
7 | 6 | adantl 485 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 = (𝐹‘𝑦)) → (𝜑 ↔ 𝜓)) |
8 | 1, 5, 7 | rexxfr2d 5277 | 1 ⊢ (𝐹 Fn 𝐴 → (∃𝑥 ∈ ran 𝐹𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 Vcvv 3441 ran crn 5520 Fn wfn 6319 ‘cfv 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-fv 6332 |
This theorem is referenced by: elrnrexdm 6832 wemapwe 9144 rexanuz 14697 climsup 15018 supcvg 15203 ruclem12 15586 prmreclem6 16247 vdwmc 16304 znunit 20255 lmbr2 21864 lmff 21906 1stcfb 22050 imasf1oxms 23096 lebnumlem3 23568 lmmbr2 23863 lmcau 23917 bcthlem4 23931 mbfsup 24268 itg2monolem1 24354 itg2gt0 24364 ostth 26223 uhgrvtxedgiedgb 26929 dfnbgr3 27128 vdn0conngrumgrv2 27981 erdszelem10 32560 neibastop2lem 33821 filnetlem4 33842 mblfinlem2 35095 istotbnd3 35209 sstotbnd 35213 heibor 35259 nacsfix 39653 fnwe2lem2 39995 climinf 42248 |
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