Theorem ov 5844

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 5731	. . . . 5 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
2		ov.6	. . . . . 6 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}
3	2	fveq1i 5376	. . . . 5 ⊢ (𝐹‘⟨𝐴, 𝐵⟩) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩)
4	1, 3	eqtri 2135	. . . 4 ⊢ (𝐴𝐹𝐵) = ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩)
5	4	eqeq1i 2122	. . 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 5828	. . . . 5 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} Fn {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)}
8		eleq1 2177	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑅 ↔ 𝐴 ∈ 𝑅))
9	8	anbi1d 458	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ↔ (𝐴 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)))
10		eleq1 2177	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑆 ↔ 𝐵 ∈ 𝑆))
11	10	anbi2d 457	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ↔ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)))
12	9, 11	opelopabg 4150	. . . . . 6 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)} ↔ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)))
13	12	ibir 176	. . . . 5 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)})
14		fnopfvb 5417	. . . . 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 408	. . . 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 462	. . . . . 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 462	. . . . . 6 ⊢ (𝑦 = 𝐵 → (((𝐴 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜓) ↔ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝜒)))
21		ov.4	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
22	21	anbi2d 457	. . . . . 6 ⊢ (𝑧 = 𝐶 → (((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝜒) ↔ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝜃)))
23	18, 20, 22	eloprabg 5813	. . . . 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 1287	. . . 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 187	. . 3 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)}‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝜃)))
26	5, 25	syl5bb 191	. 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵) = 𝐶 ↔ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) ∧ 𝜃)))
27	26	bianabs 583	1 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵) = 𝐶 ↔ 𝜃))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1314   ∈ wcel 1463  ∃!weu 1975  Vcvv 2657  ⟨cop 3496  {copab 3948   Fn wfn 5076  ‘cfv 5081  (class class class)co 5728  {coprab 5729

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-sbc 2879  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-iota 5046  df-fun 5083  df-fn 5084  df-fv 5089  df-ov 5731  df-oprab 5732

This theorem is referenced by: (None)