Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > abrexex | GIF version |
Description: Existence of a class abstraction of existentially restricted sets. 𝑥 is normally a free-variable parameter in the class expression substituted for 𝐵, which can be thought of as 𝐵(𝑥). This simple-looking theorem is actually quite powerful and appears to involve the Axiom of Replacement in an intrinsic way, as can be seen by tracing back through the path mptexg 5653, funex 5651, fnex 5650, resfunexg 5649, and funimaexg 5215. See also abrexex2 6030. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
abrexex.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
abrexex | ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2140 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | 1 | rnmpt 4795 | . 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} |
3 | abrexex.1 | . . . 4 ⊢ 𝐴 ∈ V | |
4 | 3 | mptex 5654 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V |
5 | 4 | rnex 4814 | . 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V |
6 | 2, 5 | eqeltrri 2214 | 1 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ wcel 1481 {cab 2126 ∃wrex 2418 Vcvv 2689 ↦ cmpt 3997 ran crn 4548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 |
This theorem is referenced by: ab2rexex 6037 shftfval 10625 |
Copyright terms: Public domain | W3C validator |