Theorem ovmpordx 47005

Description: Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpodxf 7557. (Contributed by AV, 30-Mar-2019.)

Hypotheses

Ref	Expression
ovmpordx.1	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅))
ovmpordx.2	⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆)
ovmpordx.3	⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐿)
ovmpordx.4	⊢ (𝜑 → 𝐴 ∈ 𝐿)
ovmpordx.5	⊢ (𝜑 → 𝐵 ∈ 𝐷)
ovmpordx.6	⊢ (𝜑 → 𝑆 ∈ 𝑋)

Assertion

Ref	Expression
ovmpordx	⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆)

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

Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐿(𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem ovmpordx

Step	Hyp	Ref	Expression
1		ovmpordx.1	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅))
2		ovmpordx.2	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆)
3		ovmpordx.3	. 2 ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐿)
4		ovmpordx.4	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐿)
5		ovmpordx.5	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
6		ovmpordx.6	. 2 ⊢ (𝜑 → 𝑆 ∈ 𝑋)
7		nfv 1917	. 2 ⊢ Ⅎ𝑥𝜑
8		nfv 1917	. 2 ⊢ Ⅎ𝑦𝜑
9		nfcv 2903	. 2 ⊢ Ⅎ𝑦𝐴
10		nfcv 2903	. 2 ⊢ Ⅎ𝑥𝐵
11		nfcv 2903	. 2 ⊢ Ⅎ𝑥𝑆
12		nfcv 2903	. 2 ⊢ Ⅎ𝑦𝑆
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	ovmpordxf 47004	1 ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  (class class class)co 7408   ∈ cmpo 7410

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413

This theorem is referenced by:  ovmpox2  47006