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Mirrors > Home > ILE Home > Th. List > abrexex2 | GIF version |
Description: Existence of an existentially restricted class abstraction. 𝜑 is normally has free-variable parameters 𝑥 and 𝑦. See also abrexex 6096. (Contributed by NM, 12-Sep-2004.) |
Ref | Expression |
---|---|
abrexex2.1 | ⊢ 𝐴 ∈ V |
abrexex2.2 | ⊢ {𝑦 ∣ 𝜑} ∈ V |
Ref | Expression |
---|---|
abrexex2 | ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1521 | . . . 4 ⊢ Ⅎ𝑧∃𝑥 ∈ 𝐴 𝜑 | |
2 | nfcv 2312 | . . . . 5 ⊢ Ⅎ𝑦𝐴 | |
3 | nfs1v 1932 | . . . . 5 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝜑 | |
4 | 2, 3 | nfrexxy 2509 | . . . 4 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑 |
5 | sbequ12 1764 | . . . . 5 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑦]𝜑)) | |
6 | 5 | rexbidv 2471 | . . . 4 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑)) |
7 | 1, 4, 6 | cbvab 2294 | . . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑} |
8 | df-clab 2157 | . . . . 5 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜑} ↔ [𝑧 / 𝑦]𝜑) | |
9 | 8 | rexbii 2477 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑) |
10 | 9 | abbii 2286 | . . 3 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑦 ∣ 𝜑}} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 [𝑧 / 𝑦]𝜑} |
11 | 7, 10 | eqtr4i 2194 | . 2 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑦 ∣ 𝜑}} |
12 | df-iun 3875 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑦 ∣ 𝜑}} | |
13 | abrexex2.1 | . . . 4 ⊢ 𝐴 ∈ V | |
14 | abrexex2.2 | . . . 4 ⊢ {𝑦 ∣ 𝜑} ∈ V | |
15 | 13, 14 | iunex 6102 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ V |
16 | 12, 15 | eqeltrri 2244 | . 2 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑦 ∣ 𝜑}} ∈ V |
17 | 11, 16 | eqeltri 2243 | 1 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V |
Colors of variables: wff set class |
Syntax hints: [wsb 1755 ∈ wcel 2141 {cab 2156 ∃wrex 2449 Vcvv 2730 ∪ ciun 3873 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 |
This theorem is referenced by: abexssex 6104 abexex 6105 oprabrexex2 6109 ab2rexex 6110 ab2rexex2 6111 |
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