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Mirrors > Home > ILE Home > Th. List > repizf2 | GIF version |
Description: Replacement. This version of replacement is stronger than repizf 4044 in the sense that 𝜑 does not need to map all values of 𝑥 in 𝑤 to a value of 𝑦. The resulting set contains those elements for which there is a value of 𝑦 and in that sense, this theorem combines repizf 4044 with ax-sep 4046. Another variation would be ∀𝑥 ∈ 𝑤∃*𝑦𝜑 → {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑤 ∧ 𝜑)} ∈ V but we don't have a proof of that yet. (Contributed by Jim Kingdon, 7-Sep-2018.) |
Ref | Expression |
---|---|
repizf2.1 | ⊢ Ⅎ𝑧𝜑 |
Ref | Expression |
---|---|
repizf2 | ⊢ (∀𝑥 ∈ 𝑤 ∃*𝑦𝜑 → ∃𝑧∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃𝑦 ∈ 𝑧 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2689 | . . 3 ⊢ 𝑤 ∈ V | |
2 | 1 | rabex 4072 | . 2 ⊢ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} ∈ V |
3 | repizf2lem 4085 | . . . 4 ⊢ (∀𝑥 ∈ 𝑤 ∃*𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑) | |
4 | nfcv 2281 | . . . . . 6 ⊢ Ⅎ𝑥𝑣 | |
5 | nfrab1 2610 | . . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} | |
6 | 4, 5 | raleqf 2622 | . . . . 5 ⊢ (𝑣 = {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → (∀𝑥 ∈ 𝑣 ∃!𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑)) |
7 | repizf2.1 | . . . . . 6 ⊢ Ⅎ𝑧𝜑 | |
8 | 7 | repizf 4044 | . . . . 5 ⊢ (∀𝑥 ∈ 𝑣 ∃!𝑦𝜑 → ∃𝑧∀𝑥 ∈ 𝑣 ∃𝑦 ∈ 𝑧 𝜑) |
9 | 6, 8 | syl6bir 163 | . . . 4 ⊢ (𝑣 = {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → (∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑 → ∃𝑧∀𝑥 ∈ 𝑣 ∃𝑦 ∈ 𝑧 𝜑)) |
10 | 3, 9 | syl5bi 151 | . . 3 ⊢ (𝑣 = {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → (∀𝑥 ∈ 𝑤 ∃*𝑦𝜑 → ∃𝑧∀𝑥 ∈ 𝑣 ∃𝑦 ∈ 𝑧 𝜑)) |
11 | df-rab 2425 | . . . . . 6 ⊢ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} = {𝑥 ∣ (𝑥 ∈ 𝑤 ∧ ∃𝑦𝜑)} | |
12 | nfv 1508 | . . . . . . . 8 ⊢ Ⅎ𝑧 𝑥 ∈ 𝑤 | |
13 | 7 | nfex 1616 | . . . . . . . 8 ⊢ Ⅎ𝑧∃𝑦𝜑 |
14 | 12, 13 | nfan 1544 | . . . . . . 7 ⊢ Ⅎ𝑧(𝑥 ∈ 𝑤 ∧ ∃𝑦𝜑) |
15 | 14 | nfab 2286 | . . . . . 6 ⊢ Ⅎ𝑧{𝑥 ∣ (𝑥 ∈ 𝑤 ∧ ∃𝑦𝜑)} |
16 | 11, 15 | nfcxfr 2278 | . . . . 5 ⊢ Ⅎ𝑧{𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} |
17 | 16 | nfeq2 2293 | . . . 4 ⊢ Ⅎ𝑧 𝑣 = {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} |
18 | 4, 5 | raleqf 2622 | . . . 4 ⊢ (𝑣 = {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → (∀𝑥 ∈ 𝑣 ∃𝑦 ∈ 𝑧 𝜑 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃𝑦 ∈ 𝑧 𝜑)) |
19 | 17, 18 | exbid 1595 | . . 3 ⊢ (𝑣 = {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → (∃𝑧∀𝑥 ∈ 𝑣 ∃𝑦 ∈ 𝑧 𝜑 ↔ ∃𝑧∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃𝑦 ∈ 𝑧 𝜑)) |
20 | 10, 19 | sylibd 148 | . 2 ⊢ (𝑣 = {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → (∀𝑥 ∈ 𝑤 ∃*𝑦𝜑 → ∃𝑧∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃𝑦 ∈ 𝑧 𝜑)) |
21 | 2, 20 | vtocle 2760 | 1 ⊢ (∀𝑥 ∈ 𝑤 ∃*𝑦𝜑 → ∃𝑧∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃𝑦 ∈ 𝑧 𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 Ⅎwnf 1436 ∃wex 1468 ∃!weu 1999 ∃*wmo 2000 {cab 2125 ∀wral 2416 ∃wrex 2417 {crab 2420 |
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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rab 2425 df-v 2688 df-in 3077 df-ss 3084 |
This theorem is referenced by: (None) |
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