Theorem ovmpordx 48528

Description: Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpodxf 7506. (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 1915	. 2 ⊢ Ⅎ𝑥𝜑
8		nfv 1915	. 2 ⊢ Ⅎ𝑦𝜑
9		nfcv 2896	. 2 ⊢ Ⅎ𝑦𝐴
10		nfcv 2896	. 2 ⊢ Ⅎ𝑥𝐵
11		nfcv 2896	. 2 ⊢ Ⅎ𝑥𝑆
12		nfcv 2896	. 2 ⊢ Ⅎ𝑦𝑆
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	ovmpordxf 48527	1 ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  (class class class)co 7356   ∈ cmpo 7358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-iota 6446  df-fun 6492  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361

This theorem is referenced by:  ovmpox2  48529