Theorem xpmapenlem1 3391

Description: Lemma for xpmapen 3396.

Hypotheses

Ref	Expression
xpmapen.1	⊢ A ∈ V
xpmapen.2	⊢ B ∈ V
xpmapen.3	⊢ C ∈ V
xpmapenlem.4	⊢ D = {⟨z, w⟩∣(z ∈ C ∧ w = ∪dom {(x ‘z)})}
xpmapenlem.5	⊢ R = {⟨z, w⟩∣(z ∈ C ∧ w = ∪ran {(x ‘z)})}
xpmapenlem.6	⊢ S = {⟨z, w⟩∣(z ∈ C ∧ w = ⟨(∪dom {y} ‘z), (∪ran {y} ‘z)⟩)}

Assertion

Ref	Expression
xpmapenlem1	⊢ ((y = ⟨D, R⟩ → ∀z y = ⟨D, R⟩) ∧ (y = ⟨D, R⟩ → ∀w y = ⟨D, R⟩))

Distinct variable group(s):   x,y,z,w,A   x,B,y,z,w   x,C,y,z,w   y,D   y,R   x,S

Proof of Theorem xpmapenlem1

Step	Hyp	Ref	Expression
1		hbopab1 2112	. . . . 5 ⊢ (y ∈ {⟨z, w⟩∣(z ∈ C ∧ w = ∪dom {(x ‘z)})} → ∀z y ∈ {⟨z, w⟩∣(z ∈ C ∧ w = ∪dom {(x ‘z)})})
2		xpmapenlem.4	. . . . . 6 ⊢ D = {⟨z, w⟩∣(z ∈ C ∧ w = ∪dom {(x ‘z)})}
3	2	eleq2i 1153	. . . . 5 ⊢ (y ∈ D ↔ y ∈ {⟨z, w⟩∣(z ∈ C ∧ w = ∪dom {(x ‘z)})})
4	3	bial 695	. . . . 5 ⊢ (∀z y ∈ D ↔ ∀z y ∈ {⟨z, w⟩∣(z ∈ C ∧ w = ∪dom {(x ‘z)})})
5	1, 3, 4	3imtr4 192	. . . 4 ⊢ (y ∈ D → ∀z y ∈ D)
6		hbopab1 2112	. . . . 5 ⊢ (y ∈ {⟨z, w⟩∣(z ∈ C ∧ w = ∪ran {(x ‘z)})} → ∀z y ∈ {⟨z, w⟩∣(z ∈ C ∧ w = ∪ran {(x ‘z)})})
7		xpmapenlem.5	. . . . . 6 ⊢ R = {⟨z, w⟩∣(z ∈ C ∧ w = ∪ran {(x ‘z)})}
8	7	eleq2i 1153	. . . . 5 ⊢ (y ∈ R ↔ y ∈ {⟨z, w⟩∣(z ∈ C ∧ w = ∪ran {(x ‘z)})})
9	8	bial 695	. . . . 5 ⊢ (∀z y ∈ R ↔ ∀z y ∈ {⟨z, w⟩∣(z ∈ C ∧ w = ∪ran {(x ‘z)})})
10	6, 8, 9	3imtr4 192	. . . 4 ⊢ (y ∈ R → ∀z y ∈ R)
11	5, 10	hbop 1879	. . 3 ⊢ (y ∈ ⟨D, R⟩ → ∀z y ∈ ⟨D, R⟩)
12	11	hbeleq 1173	. 2 ⊢ (y = ⟨D, R⟩ → ∀z y = ⟨D, R⟩)
13		hbopab2 2113	. . . . 5 ⊢ (y ∈ {⟨z, w⟩∣(z ∈ C ∧ w = ∪dom {(x ‘z)})} → ∀w y ∈ {⟨z, w⟩∣(z ∈ C ∧ w = ∪dom {(x ‘z)})})
14	3	bial 695	. . . . 5 ⊢ (∀w y ∈ D ↔ ∀w y ∈ {⟨z, w⟩∣(z ∈ C ∧ w = ∪dom {(x ‘z)})})
15	13, 3, 14	3imtr4 192	. . . 4 ⊢ (y ∈ D → ∀w y ∈ D)
16		hbopab2 2113	. . . . 5 ⊢ (y ∈ {⟨z, w⟩∣(z ∈ C ∧ w = ∪ran {(x ‘z)})} → ∀w y ∈ {⟨z, w⟩∣(z ∈ C ∧ w = ∪ran {(x ‘z)})})
17	8	bial 695	. . . . 5 ⊢ (∀w y ∈ R ↔ ∀w y ∈ {⟨z, w⟩∣(z ∈ C ∧ w = ∪ran {(x ‘z)})})
18	16, 8, 17	3imtr4 192	. . . 4 ⊢ (y ∈ R → ∀w y ∈ R)
19	15, 18	hbop 1879	. . 3 ⊢ (y ∈ ⟨D, R⟩ → ∀w y ∈ ⟨D, R⟩)
20	19	hbeleq 1173	. 2 ⊢ (y = ⟨D, R⟩ → ∀w y = ⟨D, R⟩)
21	12, 20	pm3.2i 234	1 ⊢ ((y = ⟨D, R⟩ → ∀z y = ⟨D, R⟩) ∧ (y = ⟨D, R⟩ → ∀w y = ⟨D, R⟩))

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672   = wceq 1091   ∈ wcel 1092  Vcvv 1348  {csn 1808  ⟨cop 1810  ∪cuni 1919  {copab 2055  dom cdm 2410  ran crn 2411   ‘cfv 2422

This theorem is referenced by:  xpmapenlem3 3393  xpmapenlem5 3395

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098

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