Theorem ovmpodx 7520

Description: Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.)

Hypotheses

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

Assertion

Ref	Expression
ovmpodx	⊢ (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 ovmpodx

Step	Hyp	Ref	Expression
1		ovmpodx.1	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅))
2		ovmpodx.2	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆)
3		ovmpodx.3	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐷 = 𝐿)
4		ovmpodx.4	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
5		ovmpodx.5	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐿)
6		ovmpodx.6	. 2 ⊢ (𝜑 → 𝑆 ∈ 𝑋)
7		nfv 1914	. 2 ⊢ Ⅎ𝑥𝜑
8		nfv 1914	. 2 ⊢ Ⅎ𝑦𝜑
9		nfcv 2891	. 2 ⊢ Ⅎ𝑦𝐴
10		nfcv 2891	. 2 ⊢ Ⅎ𝑥𝐵
11		nfcv 2891	. 2 ⊢ Ⅎ𝑥𝑆
12		nfcv 2891	. 2 ⊢ Ⅎ𝑦𝑆
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	ovmpodxf 7519	1 ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  (class class class)co 7369   ∈ cmpo 7371

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 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6452  df-fun 6501  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374

This theorem is referenced by:  ovmpod  7521  ovmpox  7522  dpjfval  19963  fgval  23733  om1val  24906  pi1val  24913  dvfval  25774  dvnfval  25800  taylfval  26242  line  48694  rrxline  48696