Theorem elmpocl1 7655

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.)

Hypothesis

Ref	Expression
elmpocl.f	⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Assertion

Ref	Expression
elmpocl1	⊢ (𝑋 ∈ (𝑆𝐹𝑇) → 𝑆 ∈ 𝐴)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Allowed substitution hints:   𝐶(𝑥, 𝑦)   𝑆(𝑥, 𝑦)   𝑇(𝑥, 𝑦)   𝐹(𝑥, 𝑦)   𝑋(𝑥, 𝑦)

Proof of Theorem elmpocl1

Step	Hyp	Ref	Expression
1		elmpocl.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
2	1	elmpocl 7654	. 2 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐵))
3	2	simpld 500	1 ⊢ (𝑋 ∈ (𝑆𝐹𝑇) → 𝑆 ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  (class class class)co 7412   ∈ cmpo 7414

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5657  df-dm 5661  df-iota 6487  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417

This theorem is used by:  iccssico2  13532  mhmrcl1  18962  rhmrcl1  20687  cncfrss  25192  2clwwlk2clwwlklem  30929  lbioc  46469  rellan  50675  relran  50676