Theorem ovmpox 7574

Description: The value of an operation class abstraction. Variant of ovmpoga 7575 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 3471	. 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 487	. . 3 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆)
6		ovmpox.2	. . . 4 ⊢ (𝑥 = 𝐴 → 𝐷 = 𝐿)
7	6	adantl 487	. . 3 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) ∧ 𝑥 = 𝐴) → 𝐷 = 𝐿)
8		simp1 1154	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) → 𝐴 ∈ 𝐶)
9		simp2 1155	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) → 𝐵 ∈ 𝐿)
10		simp3 1156	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) → 𝑆 ∈ V)
11	3, 5, 7, 8, 9, 10	ovmpodx 7572	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) → (𝐴𝐹𝐵) = 𝑆)
12	1, 11	syl3an3 1183	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  Vcvv 3450  (class class class)co 7421   ∈ cmpo 7423

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6541  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426

This theorem is used by:  evls1fval  22643  ptbasfi  23906  cutsval  28172  tglngval  29020  extdgval  34332  igenval  39089  isisubgr  49054  gpgov  49234  lcoop  49617