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Mirrors > Home > ILE Home > Th. List > elmpocl1 | GIF version |
Description: If a two-parameter class is inhabited, the first argument is in its nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan O'Rear, 7-Mar-2015.) |
Ref | Expression |
---|---|
elmpocl.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Ref | Expression |
---|---|
elmpocl1 | ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → 𝑆 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmpocl.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
2 | 1 | elmpocl 6020 | . 2 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐵)) |
3 | 2 | simpld 111 | 1 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → 𝑆 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1335 ∈ wcel 2128 (class class class)co 5826 ∈ cmpo 5828 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-pow 4137 ax-pr 4171 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-opab 4028 df-id 4255 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-iota 5137 df-fun 5174 df-fv 5180 df-ov 5829 df-oprab 5830 df-mpo 5831 |
This theorem is referenced by: pmresg 6623 iccssico2 9857 elfzoel1 10053 cncfrss 13032 limccl 13098 |
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