Theorem fvmptopab 7472

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 6882	. . . . . 6 ⊢ (𝑧 = 𝑍 → (𝐹‘𝑧) = (𝐹‘𝑍))
2	1	breqd 5118	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑥(𝐹‘𝑧)𝑦 ↔ 𝑥(𝐹‘𝑍)𝑦))
3		fvmptopab.1	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝜑 ↔ 𝜓))
4	2, 3	anbi12d 644	. . . 4 ⊢ (𝑧 = 𝑍 → ((𝑥(𝐹‘𝑧)𝑦 ∧ 𝜑) ↔ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)))
5	4	opabbidv 5175	. . 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 7471	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} ∈ V
8	5, 6, 7	fvmpt 6990	. 2 ⊢ (𝑍 ∈ V → (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)})
9		fvprc 6874	. . 3 ⊢ (¬ 𝑍 ∈ V → (𝑀‘𝑍) = ∅)
10		elopabran 5544	. . . . . 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 6874	. . . . 5 ⊢ (¬ 𝑍 ∈ V → (𝐹‘𝑍) = ∅)
13	11, 12	sseqtrid 3976	. . . 4 ⊢ (¬ 𝑍 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} ⊆ ∅)
14		ss0 4355	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} ⊆ ∅ → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} = ∅)
15	13, 14	syl 18	. . 3 ⊢ (¬ 𝑍 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} = ∅)
16	9, 15	eqtr4d 2800	. 2 ⊢ (¬ 𝑍 ∈ V → (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)})
17	8, 16	pm2.61i 184	1 ⊢ (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)}

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3453   ⊆ wss 3902  ∅c0 4282   class class class wbr 5107  {copab 5171   ↦ cmpt 5190  ‘cfv 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6493  df-fun 6539  df-fv 6545

This theorem is used by:  trlsfval  30165  pthsfval  30191  spthsfval  30192  clwlks  30246  crcts  30262  cycls  30263  eupths  30688