Theorem ovmpordx 47273

Description: Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpodxf 7553. (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 1909	. 2 ⊢ Ⅎ𝑥𝜑
8		nfv 1909	. 2 ⊢ Ⅎ𝑦𝜑
9		nfcv 2897	. 2 ⊢ Ⅎ𝑦𝐴
10		nfcv 2897	. 2 ⊢ Ⅎ𝑥𝐵
11		nfcv 2897	. 2 ⊢ Ⅎ𝑥𝑆
12		nfcv 2897	. 2 ⊢ Ⅎ𝑦𝑆
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	ovmpordxf 47272	1 ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  (class class class)co 7404   ∈ cmpo 7406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409

This theorem is referenced by:  ovmpox2  47274