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Mirrors > Home > ILE Home > Th. List > repizf2 | GIF version |
Description: Replacement. This version of replacement is stronger than repizf 4092 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 4092 with ax-sep 4094. 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 2724 | . . 3 ⊢ 𝑤 ∈ V | |
2 | 1 | rabex 4120 | . 2 ⊢ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} ∈ V |
3 | repizf2lem 4134 | . . . 4 ⊢ (∀𝑥 ∈ 𝑤 ∃*𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑) | |
4 | nfcv 2306 | . . . . . 6 ⊢ Ⅎ𝑥𝑣 | |
5 | nfrab1 2643 | . . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} | |
6 | 4, 5 | raleqf 2655 | . . . . 5 ⊢ (𝑣 = {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → (∀𝑥 ∈ 𝑣 ∃!𝑦𝜑 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃!𝑦𝜑)) |
7 | repizf2.1 | . . . . . 6 ⊢ Ⅎ𝑧𝜑 | |
8 | 7 | repizf 4092 | . . . . 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 2451 | . . . . . 6 ⊢ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} = {𝑥 ∣ (𝑥 ∈ 𝑤 ∧ ∃𝑦𝜑)} | |
12 | nfv 1515 | . . . . . . . 8 ⊢ Ⅎ𝑧 𝑥 ∈ 𝑤 | |
13 | 7 | nfex 1624 | . . . . . . . 8 ⊢ Ⅎ𝑧∃𝑦𝜑 |
14 | 12, 13 | nfan 1552 | . . . . . . 7 ⊢ Ⅎ𝑧(𝑥 ∈ 𝑤 ∧ ∃𝑦𝜑) |
15 | 14 | nfab 2311 | . . . . . 6 ⊢ Ⅎ𝑧{𝑥 ∣ (𝑥 ∈ 𝑤 ∧ ∃𝑦𝜑)} |
16 | 11, 15 | nfcxfr 2303 | . . . . 5 ⊢ Ⅎ𝑧{𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} |
17 | 16 | nfeq2 2318 | . . . 4 ⊢ Ⅎ𝑧 𝑣 = {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} |
18 | 4, 5 | raleqf 2655 | . . . 4 ⊢ (𝑣 = {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑} → (∀𝑥 ∈ 𝑣 ∃𝑦 ∈ 𝑧 𝜑 ↔ ∀𝑥 ∈ {𝑥 ∈ 𝑤 ∣ ∃𝑦𝜑}∃𝑦 ∈ 𝑧 𝜑)) |
19 | 17, 18 | exbid 1603 | . . 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 2795 | 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 1342 Ⅎwnf 1447 ∃wex 1479 ∃!weu 2013 ∃*wmo 2014 {cab 2150 ∀wral 2442 ∃wrex 2443 {crab 2446 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-coll 4091 ax-sep 4094 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rab 2451 df-v 2723 df-in 3117 df-ss 3124 |
This theorem is referenced by: (None) |
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