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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 5686. (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 1418 | . . . 4 ⊢ Ⅎz∃x ∈ A φ | |
2 | nfcv 2175 | . . . . 5 ⊢ ℲyA | |
3 | nfs1v 1812 | . . . . 5 ⊢ Ⅎy[z / y]φ | |
4 | 2, 3 | nfrexxy 2355 | . . . 4 ⊢ Ⅎy∃x ∈ A [z / y]φ |
5 | sbequ12 1651 | . . . . 5 ⊢ (y = z → (φ ↔ [z / y]φ)) | |
6 | 5 | rexbidv 2321 | . . . 4 ⊢ (y = z → (∃x ∈ A φ ↔ ∃x ∈ A [z / y]φ)) |
7 | 1, 4, 6 | cbvab 2157 | . . 3 ⊢ {y ∣ ∃x ∈ A φ} = {z ∣ ∃x ∈ A [z / y]φ} |
8 | df-clab 2024 | . . . . 5 ⊢ (z ∈ {y ∣ φ} ↔ [z / y]φ) | |
9 | 8 | rexbii 2325 | . . . 4 ⊢ (∃x ∈ A z ∈ {y ∣ φ} ↔ ∃x ∈ A [z / y]φ) |
10 | 9 | abbii 2150 | . . 3 ⊢ {z ∣ ∃x ∈ A z ∈ {y ∣ φ}} = {z ∣ ∃x ∈ A [z / y]φ} |
11 | 7, 10 | eqtr4i 2060 | . 2 ⊢ {y ∣ ∃x ∈ A φ} = {z ∣ ∃x ∈ A z ∈ {y ∣ φ}} |
12 | df-iun 3650 | . . 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 5692 | . . 3 ⊢ ∪ x ∈ A {y ∣ φ} ∈ V |
16 | 12, 15 | eqeltrri 2108 | . 2 ⊢ {z ∣ ∃x ∈ A z ∈ {y ∣ φ}} ∈ V |
17 | 11, 16 | eqeltri 2107 | 1 ⊢ {y ∣ ∃x ∈ A φ} ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1390 [wsb 1642 {cab 2023 ∃wrex 2301 Vcvv 2551 ∪ ciun 3648 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 |
This theorem is referenced by: abexssex 5694 abexex 5695 oprabrexex2 5699 ab2rexex 5700 ab2rexex2 5701 |
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