Theorem mpt2xopx0ov0 7624

Description: If the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, is the empty set, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)

Hypothesis

Ref	Expression
mpt2xopn0yelv.f	⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)

Assertion

Ref	Expression
mpt2xopx0ov0	⊢ (∅𝐹𝐾) = ∅

Distinct variable groups:   𝑥,𝑦   𝑥,𝐾   𝑥,𝐹

Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐹(𝑦)   𝐾(𝑦)

Proof of Theorem mpt2xopx0ov0

Step	Hyp	Ref	Expression
1		0nelxp 5389	. 2 ⊢ ¬ ∅ ∈ (V × V)
2		mpt2xopn0yelv.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)
3	2	mpt2xopxnop0 7623	. 2 ⊢ (¬ ∅ ∈ (V × V) → (∅𝐹𝐾) = ∅)
4	1, 3	ax-mp 5	1 ⊢ (∅𝐹𝐾) = ∅

Syntax hints:  ¬ wn 3   = wceq 1601   ∈ wcel 2106  Vcvv 3397  ∅c0 4140   × cxp 5353  ‘cfv 6135  (class class class)co 6922   ↦ cmpt2 6924  1^st c1st 7443

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446

This theorem is referenced by: (None)