Theorem ovmpodx 7280

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 1915	. 2 ⊢ Ⅎ𝑥𝜑
8		nfv 1915	. 2 ⊢ Ⅎ𝑦𝜑
9		nfcv 2955	. 2 ⊢ Ⅎ𝑦𝐴
10		nfcv 2955	. 2 ⊢ Ⅎ𝑥𝐵
11		nfcv 2955	. 2 ⊢ Ⅎ𝑥𝑆
12		nfcv 2955	. 2 ⊢ Ⅎ𝑦𝑆
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	ovmpodxf 7279	1 ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  (class class class)co 7135   ∈ cmpo 7137

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140

This theorem is referenced by:  ovmpod  7281  ovmpox  7282  dpjfval  19170  fgval  22475  om1val  23635  pi1val  23642  dvfval  24500  dvnfval  24525  taylfval  24954  line  45146  rrxline  45148