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Mirrors > Home > ILE Home > Th. List > ssoprab2i | GIF version |
Description: Inference of operation class abstraction subclass from implication. (Contributed by NM, 11-Nov-1995.) (Revised by David Abernethy, 19-Jun-2012.) |
Ref | Expression |
---|---|
ssoprab2i.1 | ⊢ (𝜑 → 𝜓) |
Ref | Expression |
---|---|
ssoprab2i | ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssoprab2i.1 | . . . . 5 ⊢ (𝜑 → 𝜓) | |
2 | 1 | anim2i 339 | . . . 4 ⊢ ((𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) → (𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜓)) |
3 | 2 | 2eximi 1580 | . . 3 ⊢ (∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑) → ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜓)) |
4 | 3 | ssopab2i 4194 | . 2 ⊢ {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} ⊆ {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜓)} |
5 | dfoprab2 5811 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
6 | dfoprab2 5811 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} = {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜓)} | |
7 | 4, 5, 6 | 3sstr4i 3133 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜓} |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∃wex 1468 ⊆ wss 3066 〈cop 3525 {copab 3983 {coprab 5768 |
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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-opab 3985 df-oprab 5771 |
This theorem is referenced by: (None) |
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