Theorem ovmpox 7303

Description: The value of an operation class abstraction. Variant of ovmpoga 7304 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 3512	. 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 484	. . 3 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆)
6		ovmpox.2	. . . 4 ⊢ (𝑥 = 𝐴 → 𝐷 = 𝐿)
7	6	adantl 484	. . 3 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) ∧ 𝑥 = 𝐴) → 𝐷 = 𝐿)
8		simp1 1132	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) → 𝐴 ∈ 𝐶)
9		simp2 1133	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) → 𝐵 ∈ 𝐿)
10		simp3 1134	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) → 𝑆 ∈ V)
11	3, 5, 7, 8, 9, 10	ovmpodx 7301	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) → (𝐴𝐹𝐵) = 𝑆)
12	1, 11	syl3an3 1161	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  Vcvv 3494  (class class class)co 7156   ∈ cmpo 7158

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-iota 6314  df-fun 6357  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161

This theorem is referenced by:  evls1fval  20482  ptbasfi  22189  tglngval  26337  extdgval  31044  scutval  33265  igenval  35354  lcoop  44515