Theorem fvmptopab 7401

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.) Add disjoint variable condition on 𝐹, 𝑥, 𝑦 to remove a sethood hypothesis. (Revised by SN, 13-Dec-2024.)

Hypotheses

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

Assertion

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

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

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

Proof of Theorem fvmptopab

Step	Hyp	Ref	Expression
1		fveq2 6822	. . . . . 6 ⊢ (𝑧 = 𝑍 → (𝐹‘𝑧) = (𝐹‘𝑍))
2	1	breqd 5102	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑥(𝐹‘𝑧)𝑦 ↔ 𝑥(𝐹‘𝑍)𝑦))
3		fvmptopab.1	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝜑 ↔ 𝜓))
4	2, 3	anbi12d 632	. . . 4 ⊢ (𝑧 = 𝑍 → ((𝑥(𝐹‘𝑧)𝑦 ∧ 𝜑) ↔ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)))
5	4	opabbidv 5157	. . 3 ⊢ (𝑧 = 𝑍 → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ 𝜑)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)})
6		fvmptopab.m	. . 3 ⊢ 𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ 𝜑)})
7		opabresex2 7400	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} ∈ V
8	5, 6, 7	fvmpt 6929	. 2 ⊢ (𝑍 ∈ V → (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)})
9		fvprc 6814	. . 3 ⊢ (¬ 𝑍 ∈ V → (𝑀‘𝑍) = ∅)
10		elopabran 5501	. . . . . 6 ⊢ (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} → 𝑧 ∈ (𝐹‘𝑍))
11	10	ssriv 3938	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} ⊆ (𝐹‘𝑍)
12		fvprc 6814	. . . . 5 ⊢ (¬ 𝑍 ∈ V → (𝐹‘𝑍) = ∅)
13	11, 12	sseqtrid 3977	. . . 4 ⊢ (¬ 𝑍 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} ⊆ ∅)
14		ss0 4352	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} ⊆ ∅ → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} = ∅)
15	13, 14	syl 17	. . 3 ⊢ (¬ 𝑍 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} = ∅)
16	9, 15	eqtr4d 2769	. 2 ⊢ (¬ 𝑍 ∈ V → (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)})
17	8, 16	pm2.61i 182	1 ⊢ (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)}

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ⊆ wss 3902  ∅c0 4283   class class class wbr 5091  {copab 5153   ↦ cmpt 5172  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-iota 6437  df-fun 6483  df-fv 6489

This theorem is referenced by:  trlsfval  29670  pthsfval  29695  spthsfval  29696  clwlks  29748  crcts  29764  cycls  29765  eupths  30175