Theorem fvmptopabOLD 7447

Description: Obsolete version of fvmptopab 7446 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 6861	. . . . . . . 8 ⊢ (𝑧 = 𝑍 → (𝐹‘𝑧) = (𝐹‘𝑍))
3	2	breqd 5121	. . . . . . 7 ⊢ (𝑧 = 𝑍 → (𝑥(𝐹‘𝑧)𝑦 ↔ 𝑥(𝐹‘𝑍)𝑦))
4	3	adantl 481	. . . . . 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 714	. . . . . 6 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → (𝜒 ↔ 𝜓))
7	4, 6	anbi12d 632	. . . . 5 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → ((𝑥(𝐹‘𝑧)𝑦 ∧ 𝜒) ↔ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)))
8	7	opabbidv 5176	. . . 4 ⊢ (((𝑍 ∈ V ∧ 𝜑) ∧ 𝑧 = 𝑍) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ 𝜒)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)})
9		simpl 482	. . . 4 ⊢ ((𝑍 ∈ V ∧ 𝜑) → 𝑍 ∈ V)
10		id 22	. . . . . 6 ⊢ (𝑥(𝐹‘𝑍)𝑦 → 𝑥(𝐹‘𝑍)𝑦)
11	10	gen2 1796	. . . . 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 481	. . . . 5 ⊢ ((𝑍 ∈ V ∧ 𝜑) → {⟨𝑥, 𝑦⟩ ∣ 𝑥(𝐹‘𝑍)𝑦} ∈ V)
14		opabbrex 7443	. . . . 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 6979	. . 3 ⊢ ((𝑍 ∈ V ∧ 𝜑) → (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)})
17	16	ex 412	. 2 ⊢ (𝑍 ∈ V → (𝜑 → (𝑀‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)}))
18		fvprc 6853	. . . 4 ⊢ (¬ 𝑍 ∈ V → (𝑀‘𝑍) = ∅)
19		br0 5159	. . . . . . . 8 ⊢ ¬ 𝑥∅𝑦
20		fvprc 6853	. . . . . . . . 9 ⊢ (¬ 𝑍 ∈ V → (𝐹‘𝑍) = ∅)
21	20	breqd 5121	. . . . . . . 8 ⊢ (¬ 𝑍 ∈ V → (𝑥(𝐹‘𝑍)𝑦 ↔ 𝑥∅𝑦))
22	19, 21	mtbiri 327	. . . . . . 7 ⊢ (¬ 𝑍 ∈ V → ¬ 𝑥(𝐹‘𝑍)𝑦)
23	22	intnanrd 489	. . . . . 6 ⊢ (¬ 𝑍 ∈ V → ¬ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓))
24	23	alrimivv 1928	. . . . 5 ⊢ (¬ 𝑍 ∈ V → ∀𝑥∀𝑦 ¬ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓))
25		opab0 5517	. . . . 5 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} = ∅ ↔ ∀𝑥∀𝑦 ¬ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓))
26	24, 25	sylibr 234	. . . 4 ⊢ (¬ 𝑍 ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} = ∅)
27	18, 26	eqtr4d 2768	. . 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 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ∅c0 4299   class class class wbr 5110  {copab 5172   ↦ cmpt 5191  ‘cfv 6514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522

This theorem is referenced by: (None)