Theorem mpoxopx0ov0 8170

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
mpoxopn0yelv.f	⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)

Assertion

Ref	Expression
mpoxopx0ov0	⊢ (∅𝐹𝐾) = ∅

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

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

Proof of Theorem mpoxopx0ov0

Step	Hyp	Ref	Expression
1		0nelxp 5668	. 2 ⊢ ¬ ∅ ∈ (V × V)
2		mpoxopn0yelv.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)
3	2	mpoxopxnop0 8169	. 2 ⊢ (¬ ∅ ∈ (V × V) → (∅𝐹𝐾) = ∅)
4	1, 3	ax-mp 5	1 ⊢ (∅𝐹𝐾) = ∅

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ∅c0 4287   × cxp 5632  ‘cfv 6502  (class class class)co 7370   ∈ cmpo 7372  1^st c1st 7943

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375  df-1st 7945  df-2nd 7946

This theorem is referenced by: (None)