Theorem fvmptopab 6697

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 condition 𝜓. (Contributed by AV, 31-Jan-2021.) (Revised by AV, 29-Oct-2021.)

Hypotheses

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

Assertion

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

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

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

Proof of Theorem fvmptopab

Step	Hyp	Ref	Expression
1		fvmptopab.3	. . . . 5 ⊢ 𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ 𝜒)})
2	1	a1i 11	. . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → 𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ 𝜒)}))
3		fveq2 6191	. . . . . . . 8 ⊢ (𝑧 = 𝑍 → (𝐹‘𝑧) = (𝐹‘𝑍))
4	3	breqd 4664	. . . . . . 7 ⊢ (𝑧 = 𝑍 → (𝑥(𝐹‘𝑧)𝑦 ↔ 𝑥(𝐹‘𝑍)𝑦))
5	4	adantl 482	. . . . . 6 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → (𝑥(𝐹‘𝑧)𝑦 ↔ 𝑥(𝐹‘𝑍)𝑦))
6		fvmptopab.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝜒 ↔ 𝜓))
7	6	adantll 750	. . . . . 6 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → (𝜒 ↔ 𝜓))
8	5, 7	anbi12d 747	. . . . 5 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → ((𝑥(𝐹‘𝑧)𝑦 ∧ 𝜒) ↔ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)))
9	8	opabbidv 4716	. . . 4 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ 𝜒)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)})
10		simpl 473	. . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → 𝑍 ∈ V)
11		id 22	. . . . . 6 ⊢ (𝑥(𝐹‘𝑍)𝑦 → 𝑥(𝐹‘𝑍)𝑦)
12	11	gen2 1723	. . . . 5 ⊢ ∀𝑥∀𝑦(𝑥(𝐹‘𝑍)𝑦 → 𝑥(𝐹‘𝑍)𝑦)
13		fvmptopab.2	. . . . . 6 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹‘𝑍)𝑦} ∈ V)
14	13	adantl 482	. . . . 5 ⊢ ((𝑍 ∈ V ∧ 𝜑) → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹‘𝑍)𝑦} ∈ V)
15		opabbrex 6695	. . . . 5 ⊢ ((∀𝑥∀𝑦(𝑥(𝐹‘𝑍)𝑦 → 𝑥(𝐹‘𝑍)𝑦) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹‘𝑍)𝑦} ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} ∈ V)
16	12, 14, 15	sylancr 695	. . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} ∈ V)
17	2, 9, 10, 16	fvmptd 6288	. . 3 ⊢ ((𝑍 ∈ V ∧ 𝜑) → (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)})
18	17	ex 450	. 2 ⊢ (𝑍 ∈ V → (𝜑 → (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)}))
19		fvprc 6185	. . . 4 ⊢ (¬ 𝑍 ∈ V → (𝑀‘𝑍) = ∅)
20		br0 4701	. . . . . . . 8 ⊢ ¬ 𝑥∅𝑦
21		fvprc 6185	. . . . . . . . 9 ⊢ (¬ 𝑍 ∈ V → (𝐹‘𝑍) = ∅)
22	21	breqd 4664	. . . . . . . 8 ⊢ (¬ 𝑍 ∈ V → (𝑥(𝐹‘𝑍)𝑦 ↔ 𝑥∅𝑦))
23	20, 22	mtbiri 317	. . . . . . 7 ⊢ (¬ 𝑍 ∈ V → ¬ 𝑥(𝐹‘𝑍)𝑦)
24	23	intnanrd 963	. . . . . 6 ⊢ (¬ 𝑍 ∈ V → ¬ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓))
25	24	alrimivv 1856	. . . . 5 ⊢ (¬ 𝑍 ∈ V → ∀𝑥∀𝑦 ¬ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓))
26		opab0 5007	. . . . 5 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} = ∅ ↔ ∀𝑥∀𝑦 ¬ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓))
27	25, 26	sylibr 224	. . . 4 ⊢ (¬ 𝑍 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} = ∅)
28	19, 27	eqtr4d 2659	. . 3 ⊢ (¬ 𝑍 ∈ V → (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)})
29	28	a1d 25	. 2 ⊢ (¬ 𝑍 ∈ V → (𝜑 → (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)}))
30	18, 29	pm2.61i 176	1 ⊢ (𝜑 → (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ∅c0 3915   class class class wbr 4653  {copab 4712   ↦ cmpt 4729  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  trlsfval  26592  pthsfval  26617  spthsfval  26618  clwlks  26668  crcts  26683  cycls  26684  eupths  27060