Theorem ov 7565

Description: The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Hypotheses

Ref	Expression
ov.1	⊢ 𝐶 ∈ V
ov.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
ov.3	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
ov.4	⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
ov.5	⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃!𝑧𝜑)
ov.6	⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}

Assertion

Ref	Expression
ov	⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵) = 𝐶 ↔ 𝜃))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝜃,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧)   𝜓(𝑥, 𝑦, 𝑧)   𝜒(𝑥, 𝑦, 𝑧)   𝐹(𝑥, 𝑦, 𝑧)

Proof of Theorem ov

Step	Hyp	Ref	Expression
1		df-ov 7424	. . . . 5 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2		ov.6	. . . . . 6 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}
3	2	fveq1i 6887	. . . . 5 ⊢ (𝐹‘⟨𝐴, 𝐵⟩) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩)
4	1, 3	eqtri 2783	. . . 4 ⊢ (𝐴𝐹𝐵) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩)
5	4	eqeq1i 2765	. . 3 ⊢ ((𝐴𝐹𝐵) = 𝐶 ↔ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝐶)
6		ov.5	. . . . . 6 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃!𝑧𝜑)
7	6	fnoprab 7546	. . . . 5 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)}
8		eleq1 2848	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑅 ↔ 𝐴 ∈ 𝑅))
9	8	anbi1d 643	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ↔ (𝐴 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)))
10		eleq1 2848	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑆 ↔ 𝐵 ∈ 𝑆))
11	10	anbi2d 642	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ↔ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)))
12	9, 11	opelopabg 5512	. . . . . 6 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)} ↔ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)))
13	12	ibir 271	. . . . 5 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)})
14		fnopfvb 6937	. . . . 5 ⊢ (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)} ∧ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)}) → (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}))
15	7, 13, 14	sylancr 599	. . . 4 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}))
16		ov.1	. . . . 5 ⊢ 𝐶 ∈ V
17		ov.2	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
18	9, 17	anbi12d 644	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑) ↔ ((𝐴 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜓)))
19		ov.3	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
20	11, 19	anbi12d 644	. . . . . 6 ⊢ (𝑦 = 𝐵 → (((𝐴 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜓) ↔ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝜒)))
21		ov.4	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
22	21	anbi2d 642	. . . . . 6 ⊢ (𝑧 = 𝐶 → (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝜒) ↔ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝜃)))
23	18, 20, 22	eloprabg 7531	. . . . 5 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ V) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} ↔ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝜃)))
24	16, 23	mp3an3 1479	. . . 4 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} ↔ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝜃)))
25	15, 24	bitrd 282	. . 3 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝜃)))
26	5, 25	bitrid 286	. 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵) = 𝐶 ↔ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝜃)))
27	26	bianabs 551	1 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵) = 𝐶 ↔ 𝜃))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃!weu 2593  Vcvv 3450  ⟨cop 4834  {copab 5166   Fn wfn 6534  ‘cfv 6539  (class class class)co 7421  {coprab 7422

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-nul 5259  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-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  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-fn 6542  df-fv 6547  df-ov 7424  df-oprab 7425

This theorem is used by: (None)