Theorem xpmapenlem3 3393

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
xpmapenlem3	⊢ ((x:C–→(A × B) ∧ y = ⟨D, R⟩) → x = S)

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 xpmapenlem3

Step	Hyp	Ref	Expression
1		ffn 2752	. . . 4 ⊢ (x:C–→(A × B) → x Fn C)
2		fnopabfv 2858	. . . 4 ⊢ (x Fn C ↔ x = {⟨z, w⟩∣(z ∈ C ∧ w = (x ‘z))})
3	1, 2	sylib 173	. . 3 ⊢ (x:C–→(A × B) → x = {⟨z, w⟩∣(z ∈ C ∧ w = (x ‘z))})
4	3	adantr 306	. 2 ⊢ ((x:C–→(A × B) ∧ y = ⟨D, R⟩) → x = {⟨z, w⟩∣(z ∈ C ∧ w = (x ‘z))})
5		ax-17 925	. . . . 5 ⊢ (x:C–→(A × B) → ∀z x:C–→(A × B))
6		xpmapen.1	. . . . . . 7 ⊢ A ∈ V
7		xpmapen.2	. . . . . . 7 ⊢ B ∈ V
8		xpmapen.3	. . . . . . 7 ⊢ C ∈ V
9		xpmapenlem.4	. . . . . . 7 ⊢ D = {⟨z, w⟩∣(z ∈ C ∧ w = ∪dom {(x ‘z)})}
10		xpmapenlem.5	. . . . . . 7 ⊢ R = {⟨z, w⟩∣(z ∈ C ∧ w = ∪ran {(x ‘z)})}
11		xpmapenlem.6	. . . . . . 7 ⊢ S = {⟨z, w⟩∣(z ∈ C ∧ w = ⟨(∪dom {y} ‘z), (∪ran {y} ‘z)⟩)}
12	6, 7, 8, 9, 10, 11	xpmapenlem1 3391	. . . . . 6 ⊢ ((y = ⟨D, R⟩ → ∀z y = ⟨D, R⟩) ∧ (y = ⟨D, R⟩ → ∀w y = ⟨D, R⟩))
13	12	pm3.26i 257	. . . . 5 ⊢ (y = ⟨D, R⟩ → ∀z y = ⟨D, R⟩)
14	5, 13	hban 704	. . . 4 ⊢ ((x:C–→(A × B) ∧ y = ⟨D, R⟩) → ∀z(x:C–→(A × B) ∧ y = ⟨D, R⟩))
15		ax-17 925	. . . . 5 ⊢ (x:C–→(A × B) → ∀w x:C–→(A × B))
16	12	pm3.27i 261	. . . . 5 ⊢ (y = ⟨D, R⟩ → ∀w y = ⟨D, R⟩)
17	15, 16	hban 704	. . . 4 ⊢ ((x:C–→(A × B) ∧ y = ⟨D, R⟩) → ∀w(x:C–→(A × B) ∧ y = ⟨D, R⟩))
18		ffvrn 2890	. . . . . . . . . 10 ⊢ ((x:C–→(A × B) ∧ z ∈ C) → (x ‘z) ∈ (A × B))
19		elxp4 2640	. . . . . . . . . . 11 ⊢ ((x ‘z) ∈ (A × B) ↔ ((x ‘z) = ⟨∪dom {(x ‘z)}, ∪ran {(x ‘z)}⟩ ∧ (∪dom {(x ‘z)} ∈ A ∧ ∪ran {(x ‘z)} ∈ B)))
20	19	pm3.26bd 259	. . . . . . . . . 10 ⊢ ((x ‘z) ∈ (A × B) → (x ‘z) = ⟨∪dom {(x ‘z)}, ∪ran {(x ‘z)}⟩)
21	18, 20	syl 12	. . . . . . . . 9 ⊢ ((x:C–→(A × B) ∧ z ∈ C) → (x ‘z) = ⟨∪dom {(x ‘z)}, ∪ran {(x ‘z)}⟩)
22	21	adantlr 310	. . . . . . . 8 ⊢ (((x:C–→(A × B) ∧ y = ⟨D, R⟩) ∧ z ∈ C) → (x ‘z) = ⟨∪dom {(x ‘z)}, ∪ran {(x ‘z)}⟩)
23	6, 6, 8, 9, 10, 11	xpmapenlem2 3392	. . . . . . . . . 10 ⊢ ((y = ⟨D, R⟩ ∧ z ∈ C) → ((∪dom {y} ‘z) = ∪dom {(x ‘z)} ∧ (∪ran {y} ‘z) = ∪ran {(x ‘z)}))
24		opeq12 1878	. . . . . . . . . 10 ⊢ (((∪dom {y} ‘z) = ∪dom {(x ‘z)} ∧ (∪ran {y} ‘z) = ∪ran {(x ‘z)}) → ⟨(∪dom {y} ‘z), (∪ran {y} ‘z)⟩ = ⟨∪dom {(x ‘z)}, ∪ran {(x ‘z)}⟩)
25	23, 24	syl 12	. . . . . . . . 9 ⊢ ((y = ⟨D, R⟩ ∧ z ∈ C) → ⟨(∪dom {y} ‘z), (∪ran {y} ‘z)⟩ = ⟨∪dom {(x ‘z)}, ∪ran {(x ‘z)}⟩)
26	25	adantll 309	. . . . . . . 8 ⊢ (((x:C–→(A × B) ∧ y = ⟨D, R⟩) ∧ z ∈ C) → ⟨(∪dom {y} ‘z), (∪ran {y} ‘z)⟩ = ⟨∪dom {(x ‘z)}, ∪ran {(x ‘z)}⟩)
27	22, 26	eqtr4d 1131	. . . . . . 7 ⊢ (((x:C–→(A × B) ∧ y = ⟨D, R⟩) ∧ z ∈ C) → (x ‘z) = ⟨(∪dom {y} ‘z), (∪ran {y} ‘z)⟩)
28	27	cleq2d 1112	. . . . . 6 ⊢ (((x:C–→(A × B) ∧ y = ⟨D, R⟩) ∧ z ∈ C) → (w = (x ‘z) ↔ w = ⟨(∪dom {y} ‘z), (∪ran {y} ‘z)⟩))
29	28	exp 291	. . . . 5 ⊢ ((x:C–→(A × B) ∧ y = ⟨D, R⟩) → (z ∈ C → (w = (x ‘z) ↔ w = ⟨(∪dom {y} ‘z), (∪ran {y} ‘z)⟩)))
30	29	pm5.32d 491	. . . 4 ⊢ ((x:C–→(A × B) ∧ y = ⟨D, R⟩) → ((z ∈ C ∧ w = (x ‘z)) ↔ (z ∈ C ∧ w = ⟨(∪dom {y} ‘z), (∪ran {y} ‘z)⟩)))
31	14, 17, 30	biopabd 2101	. . 3 ⊢ ((x:C–→(A × B) ∧ y = ⟨D, R⟩) → {⟨z, w⟩∣(z ∈ C ∧ w = (x ‘z))} = {⟨z, w⟩∣(z ∈ C ∧ w = ⟨(∪dom {y} ‘z), (∪ran {y} ‘z)⟩)})
32	31, 11	syl6eqr 1142	. 2 ⊢ ((x:C–→(A × B) ∧ y = ⟨D, R⟩) → {⟨z, w⟩∣(z ∈ C ∧ w = (x ‘z))} = S)
33	4, 32	eqtrd 1128	1 ⊢ ((x:C–→(A × B) ∧ y = ⟨D, R⟩) → x = S)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   = wceq 1091   ∈ wcel 1092  Vcvv 1348  {csn 1808  ⟨cop 1810  ∪cuni 1919  {copab 2055   × cxp 2408  dom cdm 2410  ran crn 2411   Fn wfn 2417  –→wf 2418   ‘cfv 2422

This theorem is referenced by:  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-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-fv 2438

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