Theorem elovmpt3rab 7218

Description: Implications for the value of an operation defined by the maps-to notation with a class abstration as a result having an element. (Contributed by AV, 17-Jul-2018.) (Revised by AV, 16-May-2019.)

Hypothesis

Ref	Expression
elovmpt3rab.o	⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧 ∈ 𝑀 ↦ {𝑎 ∈ 𝑁 ∣ 𝜑}))

Assertion

Ref	Expression
elovmpt3rab	⊢ ((𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑇) → (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑍 ∈ 𝑀 ∧ 𝐴 ∈ 𝑁))))

Distinct variable groups:   𝐴,𝑎   𝑀,𝑎,𝑥,𝑦,𝑧   𝑁,𝑎,𝑥,𝑦,𝑧   𝑧,𝑇   𝑥,𝑈,𝑦,𝑧   𝑋,𝑎,𝑥,𝑦,𝑧   𝑌,𝑎,𝑥,𝑦,𝑧   𝑍,𝑎,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑎)   𝐴(𝑥,𝑦,𝑧)   𝑇(𝑥,𝑦,𝑎)   𝑈(𝑎)   𝑂(𝑥,𝑦,𝑧,𝑎)   𝑍(𝑥,𝑦)

Proof of Theorem elovmpt3rab

Step	Hyp	Ref	Expression
1		elovmpt3rab.o	. 2 ⊢ 𝑂 = (𝑥 ∈ V, 𝑦 ∈ V ↦ (𝑧 ∈ 𝑀 ↦ {𝑎 ∈ 𝑁 ∣ 𝜑}))
2		eqidd 2773	. 2 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑀 = 𝑀)
3		eqidd 2773	. 2 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑁 = 𝑁)
4	1, 2, 3	elovmpt3rab1 7217	1 ⊢ ((𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑇) → (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑍 ∈ 𝑀 ∧ 𝐴 ∈ 𝑁))))

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2050  {crab 3086  Vcvv 3409   ↦ cmpt 5002  ‘cfv 6182  (class class class)co 6970   ∈ cmpo 6972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5306  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975

This theorem is referenced by: (None)