Theorem elovmpt3rab 7652

Description: Implications for the value of an operation defined by the maps-to notation with a class abstraction 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 2731	. 2 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑀 = 𝑀)
3		eqidd 2731	. 2 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑁 = 𝑁)
4	1, 2, 3	elovmpt3rab1 7651	1 ⊢ ((𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑇) → (𝐴 ∈ ((𝑋𝑂𝑌)‘𝑍) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ (𝑍 ∈ 𝑀 ∧ 𝐴 ∈ 𝑁))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3408  Vcvv 3450   ↦ cmpt 5190  ‘cfv 6513  (class class class)co 7389   ∈ cmpo 7391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-ov 7392  df-oprab 7393  df-mpo 7394

This theorem is referenced by: (None)