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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 5638, funex 5636, fnex 5635, resfunexg 5634, and funimaexg 5202. See also abrexex2 6015. (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 2137 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | 1 | rnmpt 4782 | . 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} |
3 | abrexex.1 | . . . 4 ⊢ 𝐴 ∈ V | |
4 | 3 | mptex 5639 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V |
5 | 4 | rnex 4801 | . 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V |
6 | 2, 5 | eqeltrri 2211 | 1 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 {cab 2123 ∃wrex 2415 Vcvv 2681 ↦ cmpt 3984 ran crn 4535 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 |
This theorem is referenced by: ab2rexex 6022 shftfval 10586 |
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