Theorem fvmptopab 7487

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 6907	. . . . . 6 ⊢ (𝑧 = 𝑍 → (𝐹‘𝑧) = (𝐹‘𝑍))
2	1	breqd 5159	. . . . 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 5214	. . 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 7485	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} ∈ V
8	5, 6, 7	fvmpt 7016	. 2 ⊢ (𝑍 ∈ V → (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)})
9		fvprc 6899	. . 3 ⊢ (¬ 𝑍 ∈ V → (𝑀‘𝑍) = ∅)
10		elopabran 5572	. . . . . 6 ⊢ (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} → 𝑧 ∈ (𝐹‘𝑍))
11	10	ssriv 3999	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} ⊆ (𝐹‘𝑍)
12		fvprc 6899	. . . . 5 ⊢ (¬ 𝑍 ∈ V → (𝐹‘𝑍) = ∅)
13	11, 12	sseqtrid 4048	. . . 4 ⊢ (¬ 𝑍 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} ⊆ ∅)
14		ss0 4408	. . . 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 2778	. 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 1537   ∈ wcel 2106  Vcvv 3478   ⊆ wss 3963  ∅c0 4339   class class class wbr 5148  {copab 5210   ↦ cmpt 5231  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571

This theorem is referenced by:  trlsfval  29728  pthsfval  29754  spthsfval  29755  clwlks  29805  crcts  29821  cycls  29822  eupths  30229