Theorem fvopab3g 5752

Description: Value of a function given by ordered-pair class abstraction. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)

Hypotheses

Ref	Expression
fvopab3g.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
fvopab3g.3	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
fvopab3g.4	⊢ (𝑥 ∈ 𝐶 → ∃!𝑦𝜑)
fvopab3g.5	⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}

Assertion

Ref	Expression
fvopab3g	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((𝐹‘𝐴) = 𝐵 ↔ 𝜒))

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 fvopab3g

Step	Hyp	Ref	Expression
1		eleq1 2297	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶))
2		fvopab3g.2	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	1, 2	anbi12d 473	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐶 ∧ 𝜑) ↔ (𝐴 ∈ 𝐶 ∧ 𝜓)))
4		fvopab3g.3	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
5	4	anbi2d 464	. . 3 ⊢ (𝑦 = 𝐵 → ((𝐴 ∈ 𝐶 ∧ 𝜓) ↔ (𝐴 ∈ 𝐶 ∧ 𝜒)))
6	3, 5	opelopabg 4388	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)} ↔ (𝐴 ∈ 𝐶 ∧ 𝜒)))
7		fvopab3g.4	. . . . . 6 ⊢ (𝑥 ∈ 𝐶 → ∃!𝑦𝜑)
8		fvopab3g.5	. . . . . 6 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}
9	7, 8	fnopab 5485	. . . . 5 ⊢ 𝐹 Fn 𝐶
10		fnopfvb 5718	. . . . 5 ⊢ ((𝐹 Fn 𝐶 ∧ 𝐴 ∈ 𝐶) → ((𝐹‘𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
11	9, 10	mpan 424	. . . 4 ⊢ (𝐴 ∈ 𝐶 → ((𝐹‘𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
12	8	eleq2i 2301	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐹 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)})
13	11, 12	bitrdi 196	. . 3 ⊢ (𝐴 ∈ 𝐶 → ((𝐹‘𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}))
14	13	adantr 276	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((𝐹‘𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}))
15		ibar 301	. . 3 ⊢ (𝐴 ∈ 𝐶 → (𝜒 ↔ (𝐴 ∈ 𝐶 ∧ 𝜒)))
16	15	adantr 276	. 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝜒 ↔ (𝐴 ∈ 𝐶 ∧ 𝜒)))
17	6, 14, 16	3bitr4d 220	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((𝐹‘𝐴) = 𝐵 ↔ 𝜒))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃!weu 2082   ∈ wcel 2205  ⟨cop 3694  {copab 4172   Fn wfn 5349  ‘cfv 5354

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362

This theorem is referenced by:  recmulnqg  7711