Theorem ovidi 7562

Description: The value of an operation class abstraction (weak version). (Contributed by Mario Carneiro, 29-Dec-2014.)

Hypotheses

Ref	Expression
ovidi.2	⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃*𝑧𝜑)
ovidi.3	⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}

Assertion

Ref	Expression
ovidi	⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → (𝜑 → (𝑥𝐹𝑦) = 𝑧))

Distinct variable groups:   𝑥,𝑦,𝑧   𝑧,𝑅   𝑧,𝑆

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧)   𝑅(𝑥, 𝑦)   𝑆(𝑥, 𝑦)   𝐹(𝑥, 𝑦, 𝑧)

Proof of Theorem ovidi

Step	Hyp	Ref	Expression
1		ovidi.2	. . . 4 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃*𝑧𝜑)
2		moanimv 2649	. . . 4 ⊢ (∃*𝑧((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑) ↔ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃*𝑧𝜑))
3	1, 2	mpbir 234	. . 3 ⊢ ∃*𝑧((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)
4		ovidi.3	. . 3 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}
5	3, 4	ovidig 7561	. 2 ⊢ (((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑) → (𝑥𝐹𝑦) = 𝑧)
6	5	ex 418	1 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → (𝜑 → (𝑥𝐹𝑦) = 𝑧))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃*wmo 2567  (class class class)co 7419  {coprab 7420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7422  df-oprab 7423

This theorem is used by:  ovmpt4g  7566  ov3  7582