Theorem ovmpodx 7584

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 1912	. 2 ⊢ Ⅎ𝑥𝜑
8		nfv 1912	. 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	ovmpodxf 7583	1 ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  (class class class)co 7431   ∈ cmpo 7433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436

This theorem is referenced by:  ovmpod  7585  ovmpox  7586  dpjfval  20090  fgval  23894  om1val  25077  pi1val  25084  dvfval  25947  dvnfval  25973  taylfval  26415  line  48582  rrxline  48584