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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 6920 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ V) | |
| 2 | fvelrnb 6968 | . . 3 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝑥)) | |
| 3 | eqcom 2743 | . . . 4 ⊢ ((𝐹‘𝑦) = 𝑥 ↔ 𝑥 = (𝐹‘𝑦)) | |
| 4 | 3 | rexbii 3093 | . . 3 ⊢ (∃𝑦 ∈ 𝐴 (𝐹‘𝑦) = 𝑥 ↔ ∃𝑦 ∈ 𝐴 𝑥 = (𝐹‘𝑦)) | 
| 5 | 2, 4 | bitrdi 287 | . 2 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝐴 𝑥 = (𝐹‘𝑦))) | 
| 6 | rexrn.1 | . . 3 ⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓)) | |
| 7 | 6 | adantl 481 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 = (𝐹‘𝑦)) → (𝜑 ↔ 𝜓)) | 
| 8 | 1, 5, 7 | rexxfr2d 5410 | 1 ⊢ (𝐹 Fn 𝐴 → (∃𝑥 ∈ ran 𝐹𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∃wrex 3069 Vcvv 3479 ran crn 5685 Fn wfn 6555 ‘cfv 6560 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6513 df-fun 6562 df-fn 6563 df-fv 6568 | 
| This theorem is referenced by: elrnrexdm 7108 wemapwe 9738 rexanuz 15385 climsup 15707 supcvg 15893 ruclem12 16278 prmreclem6 16960 vdwmc 17017 znunit 21583 lmbr2 23268 lmff 23310 1stcfb 23454 imasf1oxms 24503 lebnumlem3 24996 lmmbr2 25294 lmcau 25348 bcthlem4 25362 mbfsup 25700 itg2monolem1 25786 itg2gt0 25796 ostth 27684 uhgrvtxedgiedgb 29154 dfnbgr3 29356 vdn0conngrumgrv2 30216 erdszelem10 35206 neibastop2lem 36362 filnetlem4 36383 mblfinlem2 37666 istotbnd3 37779 sstotbnd 37783 heibor 37829 nacsfix 42728 fnwe2lem2 43068 climinf 45626 dfclnbgr3 47818 | 
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