Theorem oprabex2g 4026

Description: Existence of an operation class abstraction (special case).

Hypothesis

Ref	Expression
oprabex2g.1	⊢ F = {⟨⟨x, y⟩, z⟩∣((x ∈ A ⋀ y ∈ B) ⋀ z = C)}

Assertion

Ref	Expression
oprabex2g	⊢ ((A ∈ R ⋀ B ∈ S) → F ∈ V)

Distinct variable groups:   x,y,z,A   x,B,y,z   z,C

Proof of Theorem oprabex2g

Step	Hyp	Ref	Expression
1		ssexg 2726	. . . 4 ⊢ ((dom {⟨⟨x, y⟩, z⟩∣((x ∈ A ⋀ y ∈ B) ⋀ z = C)} ⊆ (A × B) ⋀ (A × B) ∈ V) → dom {⟨⟨x, y⟩, z⟩∣((x ∈ A ⋀ y ∈ B) ⋀ z = C)} ∈ V)
2		dmoprabss 4009	. . . . 5 ⊢ dom {⟨⟨x, y⟩, z⟩∣((x ∈ A ⋀ y ∈ B) ⋀ z = C)} ⊆ (A × B)
3	2	a1i 8	. . . 4 ⊢ ((A ∈ R ⋀ B ∈ S) → dom {⟨⟨x, y⟩, z⟩∣((x ∈ A ⋀ y ∈ B) ⋀ z = C)} ⊆ (A × B))
4		xpexg 3265	. . . 4 ⊢ ((A ∈ R ⋀ B ∈ S) → (A × B) ∈ V)
5	1, 3, 4	sylanc 473	. . 3 ⊢ ((A ∈ R ⋀ B ∈ S) → dom {⟨⟨x, y⟩, z⟩∣((x ∈ A ⋀ y ∈ B) ⋀ z = C)} ∈ V)
6		moeq 1923	. . . . . 6 ⊢ ∃*z z = C
7	6	moani 1425	. . . . 5 ⊢ ∃*z((x ∈ A ⋀ y ∈ B) ⋀ z = C)
8	7	funoprab 4017	. . . 4 ⊢ Fun {⟨⟨x, y⟩, z⟩∣((x ∈ A ⋀ y ∈ B) ⋀ z = C)}
9		funex 3614	. . . 4 ⊢ ((Fun {⟨⟨x, y⟩, z⟩∣((x ∈ A ⋀ y ∈ B) ⋀ z = C)} ⋀ dom {⟨⟨x, y⟩, z⟩∣((x ∈ A ⋀ y ∈ B) ⋀ z = C)} ∈ V) → {⟨⟨x, y⟩, z⟩∣((x ∈ A ⋀ y ∈ B) ⋀ z = C)} ∈ V)
10	8, 9	mpan 697	. . 3 ⊢ (dom {⟨⟨x, y⟩, z⟩∣((x ∈ A ⋀ y ∈ B) ⋀ z = C)} ∈ V → {⟨⟨x, y⟩, z⟩∣((x ∈ A ⋀ y ∈ B) ⋀ z = C)} ∈ V)
11	5, 10	syl 10	. 2 ⊢ ((A ∈ R ⋀ B ∈ S) → {⟨⟨x, y⟩, z⟩∣((x ∈ A ⋀ y ∈ B) ⋀ z = C)} ∈ V)
12		oprabex2g.1	. 2 ⊢ F = {⟨⟨x, y⟩, z⟩∣((x ∈ A ⋀ y ∈ B) ⋀ z = C)}
13	11, 12	syl5eqel 1555	1 ⊢ ((A ∈ R ⋀ B ∈ S) → F ∈ V)

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 958   ∈ wcel 960  Vcvv 1814   ⊆ wss 2050   × cxp 3174  dom cdm 3176  Fun wfun 3182  {copab2 3970

This theorem is referenced by:  oprabex2 4027  blfval 7832  grpdivfval 8077  hmeogrp 10524

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-oprab 3972

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