Theorem fvmptopab 6931

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 6411	. . . . . . . 8 ⊢ (𝑧 = 𝑍 → (𝐹‘𝑧) = (𝐹‘𝑍))
4	3	breqd 4854	. . . . . . 7 ⊢ (𝑧 = 𝑍 → (𝑥(𝐹‘𝑧)𝑦 ↔ 𝑥(𝐹‘𝑍)𝑦))
5	4	adantl 474	. . . . . 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 706	. . . . . 6 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → (𝜒 ↔ 𝜓))
8	5, 7	anbi12d 625	. . . . 5 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → ((𝑥(𝐹‘𝑧)𝑦 ∧ 𝜒) ↔ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)))
9	8	opabbidv 4909	. . . 4 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ 𝜒)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)})
10		simpl 475	. . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → 𝑍 ∈ V)
11		id 22	. . . . . 6 ⊢ (𝑥(𝐹‘𝑍)𝑦 → 𝑥(𝐹‘𝑍)𝑦)
12	11	gen2 1892	. . . . 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 474	. . . . 5 ⊢ ((𝑍 ∈ V ∧ 𝜑) → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹‘𝑍)𝑦} ∈ V)
15		opabbrex 6929	. . . . 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 582	. . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} ∈ V)
17	2, 9, 10, 16	fvmptd 6513	. . 3 ⊢ ((𝑍 ∈ V ∧ 𝜑) → (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)})
18	17	ex 402	. 2 ⊢ (𝑍 ∈ V → (𝜑 → (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)}))
19		fvprc 6404	. . . 4 ⊢ (¬ 𝑍 ∈ V → (𝑀‘𝑍) = ∅)
20		br0 4892	. . . . . . . 8 ⊢ ¬ 𝑥∅𝑦
21		fvprc 6404	. . . . . . . . 9 ⊢ (¬ 𝑍 ∈ V → (𝐹‘𝑍) = ∅)
22	21	breqd 4854	. . . . . . . 8 ⊢ (¬ 𝑍 ∈ V → (𝑥(𝐹‘𝑍)𝑦 ↔ 𝑥∅𝑦))
23	20, 22	mtbiri 319	. . . . . . 7 ⊢ (¬ 𝑍 ∈ V → ¬ 𝑥(𝐹‘𝑍)𝑦)
24	23	intnanrd 484	. . . . . 6 ⊢ (¬ 𝑍 ∈ V → ¬ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓))
25	24	alrimivv 2024	. . . . 5 ⊢ (¬ 𝑍 ∈ V → ∀𝑥∀𝑦 ¬ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓))
26		opab0 5203	. . . . 5 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} = ∅ ↔ ∀𝑥∀𝑦 ¬ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓))
27	25, 26	sylibr 226	. . . 4 ⊢ (¬ 𝑍 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} = ∅)
28	19, 27	eqtr4d 2836	. . 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 177	1 ⊢ (𝜑 → (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 385  ∀wal 1651   = wceq 1653   ∈ wcel 2157  Vcvv 3385  ∅c0 4115   class class class wbr 4843  {copab 4905   ↦ cmpt 4922  ‘cfv 6101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-iota 6064  df-fun 6103  df-fv 6109

This theorem is referenced by:  trlsfval  26948  pthsfval  26975  spthsfval  26976  clwlks  27026  crcts  27042  cycls  27043  eupths  27544