Theorem xpmapenlem3 4504

Description: Lemma for xpmapen 4507.

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 groups:   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 3633	. . . 4 ⊢ (x:C–→(A × B) → x Fn C)
2		fnopabfv 3764	. . . 4 ⊢ (x Fn C ↔ x = {⟨z, w⟩∣(z ∈ C ⋀ w = (x ‘z))})
3	1, 2	sylib 198	. . 3 ⊢ (x:C–→(A × B) → x = {⟨z, w⟩∣(z ∈ C ⋀ w = (x ‘z))})
4	3	adantr 391	. 2 ⊢ ((x:C–→(A × B) ⋀ y = ⟨D, R⟩) → x = {⟨z, w⟩∣(z ∈ C ⋀ w = (x ‘z))})
5		ax-17 973	. . . . 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 4502	. . . . . 6 ⊢ ((y = ⟨D, R⟩ → ∀z y = ⟨D, R⟩) ⋀ (y = ⟨D, R⟩ → ∀w y = ⟨D, R⟩))
13	12	pm3.26i 320	. . . . 5 ⊢ (y = ⟨D, R⟩ → ∀z y = ⟨D, R⟩)
14	5, 13	hban 1011	. . . 4 ⊢ ((x:C–→(A × B) ⋀ y = ⟨D, R⟩) → ∀z(x:C–→(A × B) ⋀ y = ⟨D, R⟩))
15		ax-17 973	. . . . 5 ⊢ (x:C–→(A × B) → ∀w x:C–→(A × B))
16	12	pm3.27i 324	. . . . 5 ⊢ (y = ⟨D, R⟩ → ∀w y = ⟨D, R⟩)
17	15, 16	hban 1011	. . . 4 ⊢ ((x:C–→(A × B) ⋀ y = ⟨D, R⟩) → ∀w(x:C–→(A × B) ⋀ y = ⟨D, R⟩))
18		ffvelrn 3820	. . . . . . . . 9 ⊢ ((x:C–→(A × B) ⋀ z ∈ C) → (x ‘z) ∈ (A × B))
19		elxp4 3459	. . . . . . . . . 10 ⊢ ((x ‘z) ∈ (A × B) ↔ ((x ‘z) = ⟨∪dom {(x ‘z)}, ∪ran {(x ‘z)}⟩ ⋀ (∪dom {(x ‘z)} ∈ A ⋀ ∪ran {(x ‘z)} ∈ B)))
20	19	pm3.26bi 322	. . . . . . . . 9 ⊢ ((x ‘z) ∈ (A × B) → (x ‘z) = ⟨∪dom {(x ‘z)}, ∪ran {(x ‘z)}⟩)
21	18, 20	syl 10	. . . . . . . 8 ⊢ ((x:C–→(A × B) ⋀ z ∈ C) → (x ‘z) = ⟨∪dom {(x ‘z)}, ∪ran {(x ‘z)}⟩)
22	21	adantlr 395	. . . . . . 7 ⊢ (((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 4503	. . . . . . . . 9 ⊢ ((y = ⟨D, R⟩ ⋀ z ∈ C) → ((∪dom { y} ‘z) = ∪dom {(x ‘z)} ⋀ (∪ran { y} ‘z) = ∪ran {(x ‘z)}))
24		opeq12 2493	. . . . . . . . 9 ⊢ (((∪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 10	. . . . . . . 8 ⊢ ((y = ⟨D, R⟩ ⋀ z ∈ C) → ⟨(∪dom { y} ‘z), (∪ran { y} ‘z)⟩ = ⟨∪dom {(x ‘z)}, ∪ran {(x ‘z)}⟩)
26	25	adantll 394	. . . . . . 7 ⊢ (((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 1513	. . . . . 6 ⊢ (((x:C–→(A × B) ⋀ y = ⟨D, R⟩) ⋀ z ∈ C) → (x ‘z) = ⟨(∪dom { y} ‘z), (∪ran { y} ‘z)⟩)
28	27	eqeq2d 1489	. . . . 5 ⊢ (((x:C–→(A × B) ⋀ y = ⟨D, R⟩) ⋀ z ∈ C) → (w = (x ‘z) ↔ w = ⟨(∪dom { y} ‘z), (∪ran { y} ‘z)⟩))
29	28	pm5.32da 651	. . . 4 ⊢ ((x:C–→(A × B) ⋀ y = ⟨D, R⟩) → ((z ∈ C ⋀ w = (x ‘z)) ↔ (z ∈ C ⋀ w = ⟨(∪dom { y} ‘z), (∪ran { y} ‘z)⟩)))
30	14, 17, 29	opabbid 2674	. . 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)⟩)})
31	30, 11	syl6eqr 1528	. 2 ⊢ ((x:C–→(A × B) ⋀ y = ⟨D, R⟩) → {⟨z, w⟩∣(z ∈ C ⋀ w = (x ‘z))} = S)
32	4, 31	eqtrd 1510	1 ⊢ ((x:C–→(A × B) ⋀ y = ⟨D, R⟩) → x = S)

Syntax hints:   → wi 3   ⋀ wa 223  ∀wal 956   = wceq 958   ∈ wcel 960  Vcvv 1814  {csn 2413  ⟨cop 2415  ∪cuni 2507  {copab 2671   × cxp 3174  dom cdm 3176  ran crn 3177   Fn wfn 3183  –→wf 3184   ‘cfv 3188

This theorem is referenced by:  xpmapenlem5 4506

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204

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