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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 2371. (Contributed by Mario Carneiro, 20-Aug-2015.) Avoid ax-13 2371. (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 317 | . . . 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 317 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (¬ ∀𝑦 ∈ ran 𝐹 ¬ 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜒)) |
6 | dfrex2 3073 | . 2 ⊢ (∃𝑦 ∈ ran 𝐹𝜓 ↔ ¬ ∀𝑦 ∈ ran 𝐹 ¬ 𝜓) | |
7 | dfrex2 3073 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜒 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜒) | |
8 | 5, 6, 7 | 3bitr4g 313 | 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 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 ↦ 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 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 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 8342 onnseq 8343 ghmcyg 19763 pgpfac1lem2 19944 pgpfac1lem3 19946 pgpfac1lem4 19947 pptbas 22510 lly1stc 22999 txbas 23070 eltsms 23636 tsmsf1o 23648 psmetutop 24075 xrge0tsms 24349 fmcfil 24788 ellimc2 25393 limcflf 25397 xrge0tsmsd 32204 rspectopn 32842 poimirlem23 36506 poimirlem24 36507 poimirlem30 36513 cntotbnd 36659 mnurndlem1 43030 |
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