Theorem ovmpox 7507

Description: The value of an operation class abstraction. Variant of ovmpoga 7508 which does not require 𝐷 and 𝑥 to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)

Hypotheses

Ref	Expression
ovmpox.1	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆)
ovmpox.2	⊢ (𝑥 = 𝐴 → 𝐷 = 𝐿)
ovmpox.3	⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)

Assertion

Ref	Expression
ovmpox	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐿,𝑦   𝑥,𝑆,𝑦

Allowed substitution hints:   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐻(𝑥,𝑦)

Proof of Theorem ovmpox

Step	Hyp	Ref	Expression
1		elex 3458	. 2 ⊢ (𝑆 ∈ 𝐻 → 𝑆 ∈ V)
2		ovmpox.3	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
3	2	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅))
4		ovmpox.1	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆)
5	4	adantl 481	. . 3 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆)
6		ovmpox.2	. . . 4 ⊢ (𝑥 = 𝐴 → 𝐷 = 𝐿)
7	6	adantl 481	. . 3 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) ∧ 𝑥 = 𝐴) → 𝐷 = 𝐿)
8		simp1 1136	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) → 𝐴 ∈ 𝐶)
9		simp2 1137	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) → 𝐵 ∈ 𝐿)
10		simp3 1138	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) → 𝑆 ∈ V)
11	3, 5, 7, 8, 9, 10	ovmpodx 7505	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) → (𝐴𝐹𝐵) = 𝑆)
12	1, 11	syl3an3 1165	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  Vcvv 3437  (class class class)co 7354   ∈ cmpo 7356

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 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6444  df-fun 6490  df-fv 6496  df-ov 7357  df-oprab 7358  df-mpo 7359

This theorem is referenced by:  evls1fval  22237  ptbasfi  23499  scutval  27744  tglngval  28532  extdgval  33689  igenval  38124  isisubgr  47989  gpgov  48169  lcoop  48539