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Mirrors > Home > MPE Home > Th. List > elmpocl1 | Structured version Visualization version GIF version |
Description: If a two-parameter class is not empty, 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 7632 | . 2 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐵)) |
3 | 2 | simpld 495 | 1 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → 𝑆 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 (class class class)co 7394 ∈ cmpo 7396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pr 5421 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-xp 5676 df-dm 5680 df-iota 6485 df-fv 6541 df-ov 7397 df-oprab 7398 df-mpo 7399 |
This theorem is referenced by: iccssico2 13382 mhmrcl1 18653 rhmrcl1 20207 cncfrss 24338 2clwwlk2clwwlklem 29528 lbioc 44063 |
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