Theorem fvmptopabOLD 7330

Description: Obsolete version of fvmptopab 7329 as of 13-Dec-2024. (Contributed by AV, 31-Jan-2021.) (Revised by AV, 29-Oct-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
fvmptopabOLD	⊢ (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 fvmptopabOLD

Step	Hyp	Ref	Expression
1		fvmptopabOLD.3	. . . 4 ⊢ 𝑀 = (𝑧 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ 𝜒)})
2		fveq2 6774	. . . . . . . 8 ⊢ (𝑧 = 𝑍 → (𝐹‘𝑧) = (𝐹‘𝑍))
3	2	breqd 5085	. . . . . . 7 ⊢ (𝑧 = 𝑍 → (𝑥(𝐹‘𝑧)𝑦 ↔ 𝑥(𝐹‘𝑍)𝑦))
4	3	adantl 482	. . . . . 6 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → (𝑥(𝐹‘𝑧)𝑦 ↔ 𝑥(𝐹‘𝑍)𝑦))
5		fvmptopabOLD.1	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝜒 ↔ 𝜓))
6	5	adantll 711	. . . . . 6 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → (𝜒 ↔ 𝜓))
7	4, 6	anbi12d 631	. . . . 5 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → ((𝑥(𝐹‘𝑧)𝑦 ∧ 𝜒) ↔ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)))
8	7	opabbidv 5140	. . . 4 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ 𝜒)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)})
9		simpl 483	. . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → 𝑍 ∈ V)
10		id 22	. . . . . 6 ⊢ (𝑥(𝐹‘𝑍)𝑦 → 𝑥(𝐹‘𝑍)𝑦)
11	10	gen2 1799	. . . . 5 ⊢ ∀𝑥∀𝑦(𝑥(𝐹‘𝑍)𝑦 → 𝑥(𝐹‘𝑍)𝑦)
12		fvmptopabOLD.2	. . . . . 6 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹‘𝑍)𝑦} ∈ V)
13	12	adantl 482	. . . . 5 ⊢ ((𝑍 ∈ V ∧ 𝜑) → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹‘𝑍)𝑦} ∈ V)
14		opabbrex 7326	. . . . 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)
15	11, 13, 14	sylancr 587	. . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} ∈ V)
16	1, 8, 9, 15	fvmptd2 6883	. . 3 ⊢ ((𝑍 ∈ V ∧ 𝜑) → (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)})
17	16	ex 413	. 2 ⊢ (𝑍 ∈ V → (𝜑 → (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)}))
18		fvprc 6766	. . . 4 ⊢ (¬ 𝑍 ∈ V → (𝑀‘𝑍) = ∅)
19		br0 5123	. . . . . . . 8 ⊢ ¬ 𝑥∅𝑦
20		fvprc 6766	. . . . . . . . 9 ⊢ (¬ 𝑍 ∈ V → (𝐹‘𝑍) = ∅)
21	20	breqd 5085	. . . . . . . 8 ⊢ (¬ 𝑍 ∈ V → (𝑥(𝐹‘𝑍)𝑦 ↔ 𝑥∅𝑦))
22	19, 21	mtbiri 327	. . . . . . 7 ⊢ (¬ 𝑍 ∈ V → ¬ 𝑥(𝐹‘𝑍)𝑦)
23	22	intnanrd 490	. . . . . 6 ⊢ (¬ 𝑍 ∈ V → ¬ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓))
24	23	alrimivv 1931	. . . . 5 ⊢ (¬ 𝑍 ∈ V → ∀𝑥∀𝑦 ¬ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓))
25		opab0 5467	. . . . 5 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} = ∅ ↔ ∀𝑥∀𝑦 ¬ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓))
26	24, 25	sylibr 233	. . . 4 ⊢ (¬ 𝑍 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} = ∅)
27	18, 26	eqtr4d 2781	. . 3 ⊢ (¬ 𝑍 ∈ V → (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)})
28	27	a1d 25	. 2 ⊢ (¬ 𝑍 ∈ V → (𝜑 → (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)}))
29	17, 28	pm2.61i 182	1 ⊢ (𝜑 → (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ∅c0 4256   class class class wbr 5074  {copab 5136   ↦ cmpt 5157  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441

This theorem is referenced by: (None)