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| Mirrors > Home > MPE Home > Th. List > ralrn | Structured version Visualization version GIF version | ||
| Description: Restricted universal 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 | 
|---|---|
| ralrn | ⊢ (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 | ralxfr2d 5409 | 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 ∀wral 3060 ∃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: ralrnmptw 7113 ralrnmpt 7115 cbvfo 7310 isoselem 7362 indexfi 9401 ordtypelem9 9567 ordtypelem10 9568 wemapwe 9738 numacn 10090 acndom 10092 rpnnen1lem3 13022 fsequb2 14018 limsuple 15515 limsupval2 15517 climsup 15707 ruclem11 16277 ruclem12 16278 prmreclem6 16960 imasaddfnlem 17574 imasvscafn 17583 cycsubgcl 19225 ghmrn 19248 ghmnsgima 19259 pgpssslw 19633 gexex 19872 dprdfcntz 20036 znf1o 21571 frlmlbs 21818 lindfrn 21842 ptcnplem 23630 kqt0lem 23745 isr0 23746 regr1lem2 23749 uzrest 23906 tmdgsum2 24105 imasf1oxmet 24386 imasf1omet 24387 bndth 24991 evth 24992 ovolficcss 25505 ovollb2lem 25524 ovolunlem1 25533 ovoliunlem1 25538 ovoliunlem2 25539 ovoliun2 25542 ovolscalem1 25549 ovolicc1 25552 voliunlem2 25587 voliunlem3 25588 ioombl1lem4 25597 uniioovol 25615 uniioombllem2 25619 uniioombllem3 25621 uniioombllem6 25624 volsup2 25641 vitalilem3 25646 mbfsup 25700 mbfinf 25701 mbflimsup 25702 itg1ge0 25722 itg1mulc 25740 itg1climres 25750 mbfi1fseqlem4 25754 itg2seq 25778 itg2monolem1 25786 itg2mono 25789 itg2i1fseq2 25792 itg2gt0 25796 itg2cnlem1 25797 itg2cn 25799 limciun 25930 plycpn 26332 hmopidmchi 32171 hmopidmpji 32172 rge0scvg 33949 mclsax 35575 mblfinlem2 37666 ismtyhmeolem 37812 nacsfix 42728 fnwe2lem2 43068 gneispace 44152 climinf 45626 liminfval2 45788 | 
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