Theorem fvopab3ig 5671

Description: Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999.)

Hypotheses

Ref	Expression
fvopab3ig.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
fvopab3ig.2	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
fvopab3ig.3	⊢ (𝑥 ∈ 𝐶 → ∃*𝑦𝜑)
fvopab3ig.4	⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}

Assertion

Ref	Expression
fvopab3ig	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝜒 → (𝐹‘𝐴) = 𝐵))

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

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

Proof of Theorem fvopab3ig

Step	Hyp	Ref	Expression
1		eleq1 2269	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶))
2		fvopab3ig.1	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	1, 2	anbi12d 473	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐶 ∧ 𝜑) ↔ (𝐴 ∈ 𝐶 ∧ 𝜓)))
4		fvopab3ig.2	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
5	4	anbi2d 464	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝐶 ∧ 𝜓) ↔ (𝐴 ∈ 𝐶 ∧ 𝜒)))
6	3, 5	opelopabg 4327	. . . . . 6 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)} ↔ (𝐴 ∈ 𝐶 ∧ 𝜒)))
7	6	biimpar 297	. . . . 5 ⊢ (((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝜒)) → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)})
8	7	exp43 372	. . . 4 ⊢ (𝐴 ∈ 𝐶 → (𝐵 ∈ 𝐷 → (𝐴 ∈ 𝐶 → (𝜒 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}))))
9	8	pm2.43a 51	. . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐵 ∈ 𝐷 → (𝜒 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)})))
10	9	imp 124	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝜒 → ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}))
11		fvopab3ig.4	. . . 4 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}
12	11	fveq1i 5595	. . 3 ⊢ (𝐹‘𝐴) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}‘𝐴)
13		funopab 5320	. . . . 5 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)} ↔ ∀𝑥∃*𝑦(𝑥 ∈ 𝐶 ∧ 𝜑))
14		fvopab3ig.3	. . . . . 6 ⊢ (𝑥 ∈ 𝐶 → ∃*𝑦𝜑)
15		moanimv 2130	. . . . . 6 ⊢ (∃*𝑦(𝑥 ∈ 𝐶 ∧ 𝜑) ↔ (𝑥 ∈ 𝐶 → ∃*𝑦𝜑))
16	14, 15	mpbir 146	. . . . 5 ⊢ ∃*𝑦(𝑥 ∈ 𝐶 ∧ 𝜑)
17	13, 16	mpgbir 1477	. . . 4 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}
18		funopfv 5636	. . . 4 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)} → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)} → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}‘𝐴) = 𝐵))
19	17, 18	ax-mp 5	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)} → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}‘𝐴) = 𝐵)
20	12, 19	eqtrid 2251	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)} → (𝐹‘𝐴) = 𝐵)
21	10, 20	syl6 33	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝜒 → (𝐹‘𝐴) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373  ∃*wmo 2056   ∈ wcel 2177  ⟨cop 3641  {copab 4115  Fun wfun 5279  ‘cfv 5285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-br 4055  df-opab 4117  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-iota 5246  df-fun 5287  df-fv 5293

This theorem is referenced by:  fvmptg  5673  fvopab6  5694  ov6g  6102