Theorem fvmptopab2 6595

Description: The function value of a mapping 𝑀 to a restricted binary relation expressed as an ordered-pair class abstraction: The restricted binary relation is a binary relation given as value of a function 𝐹 restricted by the conditions 𝜓 and 𝜏. (Contributed by AV, 31-Jan-2021.)

Hypotheses

Ref	Expression
fvmptopab1.2	⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝜒 ↔ 𝜓))
fvmptopab1.3	⊢ (𝜑 → 𝑍 ∈ 𝑉)
fvmptopab1.4	⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹‘𝑍)𝑦} ∈ V)
fvmptopab2.1	⊢ 𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ 𝜒 ∧ 𝜃)})
fvmptopab2.2	⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝜃 ↔ 𝜏))

Assertion

Ref	Expression
fvmptopab2	⊢ (𝜑 → (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓 ∧ 𝜏)})

Distinct variable groups:   𝑧,𝐹   𝑥,𝑍,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝜓,𝑧   𝜏,𝑧

Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦,𝑧)   𝜃(𝑥,𝑦,𝑧)   𝜏(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑀(𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem fvmptopab2

Step	Hyp	Ref	Expression
1		fvmptopab1.2	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝜒 ↔ 𝜓))
2		fvmptopab2.2	. . . 4 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝜃 ↔ 𝜏))
3	1, 2	anbi12d 743	. . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → ((𝜒 ∧ 𝜃) ↔ (𝜓 ∧ 𝜏)))
4		fvmptopab1.3	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
5		fvmptopab1.4	. . 3 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹‘𝑍)𝑦} ∈ V)
6		fvmptopab2.1	. . . 4 ⊢ 𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ 𝜒 ∧ 𝜃)})
7		3anass 1035	. . . . . 6 ⊢ ((𝑥(𝐹‘𝑧)𝑦 ∧ 𝜒 ∧ 𝜃) ↔ (𝑥(𝐹‘𝑧)𝑦 ∧ (𝜒 ∧ 𝜃)))
8	7	opabbii 4649	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ 𝜒 ∧ 𝜃)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ (𝜒 ∧ 𝜃))}
9	8	mpteq2i 4669	. . . 4 ⊢ (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ 𝜒 ∧ 𝜃)}) = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ (𝜒 ∧ 𝜃))})
10	6, 9	eqtri 2632	. . 3 ⊢ 𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ (𝜒 ∧ 𝜃))})
11	3, 4, 5, 10	fvmptopab1 6594	. 2 ⊢ (𝜑 → (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ (𝜓 ∧ 𝜏))})
12		3anass 1035	. . . 4 ⊢ ((𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓 ∧ 𝜏) ↔ (𝑥(𝐹‘𝑍)𝑦 ∧ (𝜓 ∧ 𝜏)))
13	12	bicomi 213	. . 3 ⊢ ((𝑥(𝐹‘𝑍)𝑦 ∧ (𝜓 ∧ 𝜏)) ↔ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓 ∧ 𝜏))
14	13	opabbii 4649	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ (𝜓 ∧ 𝜏))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓 ∧ 𝜏)}
15	11, 14	syl6eq 2660	1 ⊢ (𝜑 → (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓 ∧ 𝜏)})

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583  {copab 4642   ↦ cmpt 4643  ‘cfv 5804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812

This theorem is referenced by:  pthsfval  40927