Theorem ovmpt2x 5649

Description: The value of an operation class abstraction. Variant of ovmpt2ga 5650 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
ovmpt2x.1	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆)
ovmpt2x.2	⊢ (𝑥 = 𝐴 → 𝐷 = 𝐿)
ovmpt2x.3	⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)

Assertion

Ref	Expression
ovmpt2x	⊢ ((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 ovmpt2x

Step	Hyp	Ref	Expression
1		elex 2610	. 2 ⊢ (𝑆 ∈ 𝐻 → 𝑆 ∈ V)
2		ovmpt2x.3	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)
3	2	a1i 9	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅))
4		ovmpt2x.1	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆)
5	4	adantl 271	. . 3 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆)
6		ovmpt2x.2	. . . 4 ⊢ (𝑥 = 𝐴 → 𝐷 = 𝐿)
7	6	adantl 271	. . 3 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) ∧ 𝑥 = 𝐴) → 𝐷 = 𝐿)
8		simp1 938	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) → 𝐴 ∈ 𝐶)
9		simp2 939	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) → 𝐵 ∈ 𝐿)
10		simp3 940	. . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) → 𝑆 ∈ V)
11	3, 5, 7, 8, 9, 10	ovmpt2dx 5647	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ V) → (𝐴𝐹𝐵) = 𝑆)
12	1, 11	syl3an3 1204	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆)

Syntax hints:   → wi 4   ∧ wa 102   ∧ w3a 919   = wceq 1284   ∈ wcel 1433  Vcvv 2601  (class class class)co 5532   ↦ cmpt2 5534

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537

This theorem is referenced by: (None)