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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 7097 with a disjoint variable condition, which does not require ax-13 2369. (Contributed by Mario Carneiro, 20-Aug-2015.) Avoid ax-13 2369. (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 7094 | . . 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 3071 | . 2 ⊢ (∃𝑦 ∈ ran 𝐹𝜓 ↔ ¬ ∀𝑦 ∈ ran 𝐹 ¬ 𝜓) | |
7 | dfrex2 3071 | . 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 1539 ∈ wcel 2104 ∀wral 3059 ∃wrex 3068 ↦ cmpt 5230 ran crn 5676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-fv 6550 |
This theorem is referenced by: onoviun 8345 onnseq 8346 ghmcyg 19805 pgpfac1lem2 19986 pgpfac1lem3 19988 pgpfac1lem4 19989 pptbas 22731 lly1stc 23220 txbas 23291 eltsms 23857 tsmsf1o 23869 psmetutop 24296 xrge0tsms 24570 fmcfil 25020 ellimc2 25626 limcflf 25630 xrge0tsmsd 32479 rspectopn 33145 poimirlem23 36814 poimirlem24 36815 poimirlem30 36821 cntotbnd 36967 mnurndlem1 43342 |
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