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Mirrors > Home > MPE Home > Th. List > rexrnmptw | Structured version Visualization version GIF version |
Description: A restricted quantifier over an image set. Version of rexrnmpt 7098 with a disjoint variable condition, which does not require ax-13 2370. (Contributed by Mario Carneiro, 20-Aug-2015.) Avoid ax-13 2370. (Revised by Gino Giotto, 26-Jan-2024.) |
Ref | Expression |
---|---|
rexrnmptw.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
rexrnmptw.2 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rexrnmptw | ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexrnmptw.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | rexrnmptw.2 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
3 | 2 | notbid 318 | . . . 4 ⊢ (𝑦 = 𝐵 → (¬ 𝜓 ↔ ¬ 𝜒)) |
4 | 1, 3 | ralrnmptw 7095 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹 ¬ 𝜓 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝜒)) |
5 | 4 | notbid 318 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (¬ ∀𝑦 ∈ ran 𝐹 ¬ 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜒)) |
6 | dfrex2 3072 | . 2 ⊢ (∃𝑦 ∈ ran 𝐹𝜓 ↔ ¬ ∀𝑦 ∈ ran 𝐹 ¬ 𝜓) | |
7 | dfrex2 3072 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜒 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜒) | |
8 | 5, 6, 7 | 3bitr4g 314 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ∃wrex 3069 ↦ cmpt 5231 ran crn 5677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-fv 6551 |
This theorem is referenced by: onoviun 8347 onnseq 8348 ghmcyg 19806 pgpfac1lem2 19987 pgpfac1lem3 19989 pgpfac1lem4 19990 pptbas 22732 lly1stc 23221 txbas 23292 eltsms 23858 tsmsf1o 23870 psmetutop 24297 xrge0tsms 24571 fmcfil 25021 ellimc2 25627 limcflf 25631 xrge0tsmsd 32480 rspectopn 33146 poimirlem23 36815 poimirlem24 36816 poimirlem30 36822 cntotbnd 36968 mnurndlem1 43343 |
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