Theorem rdglem2 3944

Description: Lemma used with the recursive definition generator. This is a trivial lemma that just changes bound variables for later use.

Assertion

Ref	Expression
rdglem2	⊢ {⟨x, y⟩∣((x = ∅ ⋀ y = A) ⋁ (¬ (x = ∅ ⋁ Lim dom x) ⋀ y = (H ‘(x ‘∪dom x))) ⋁ (Lim dom x ⋀ y = ∪ran x))} = {⟨z, y⟩∣((z = ∅ ⋀ y = A) ⋁ (¬ (z = ∅ ⋁ Lim dom z) ⋀ y = (H ‘(z ‘∪dom z))) ⋁ (Lim dom z ⋀ y = ∪ran z))}

Distinct variable groups:   x,y,z   x,A,z   x,H,z

Proof of Theorem rdglem2

Step	Hyp	Ref	Expression
1		opeq1 2491	. . . . . . 7 ⊢ (x = z → ⟨x, y⟩ = ⟨z, y⟩)
2	1	eqeq2d 1489	. . . . . 6 ⊢ (x = z → (w = ⟨x, y⟩ ↔ w = ⟨z, y⟩))
3		eqeq1 1484	. . . . . . . 8 ⊢ (x = z → (x = ∅ ↔ z = ∅))
4	3	anbi1d 619	. . . . . . 7 ⊢ (x = z → ((x = ∅ ⋀ y = A) ↔ (z = ∅ ⋀ y = A)))
5		dmeq 3317	. . . . . . . . . . 11 ⊢ (x = z → dom x = dom z)
6		limeq 2966	. . . . . . . . . . 11 ⊢ (dom x = dom z → (Lim dom x ↔ Lim dom z))
7	5, 6	syl 10	. . . . . . . . . 10 ⊢ (x = z → (Lim dom x ↔ Lim dom z))
8	3, 7	orbi12d 629	. . . . . . . . 9 ⊢ (x = z → ((x = ∅ ⋁ Lim dom x) ↔ (z = ∅ ⋁ Lim dom z)))
9	8	negbid 613	. . . . . . . 8 ⊢ (x = z → (¬ (x = ∅ ⋁ Lim dom x) ↔ ¬ (z = ∅ ⋁ Lim dom z)))
10		unieq 2514	. . . . . . . . . . . 12 ⊢ (dom x = dom z → ∪dom x = ∪dom z)
11		fveq2 3730	. . . . . . . . . . . 12 ⊢ (∪dom x = ∪dom z → (x ‘∪dom x) = (x ‘∪dom z))
12	5, 10, 11	3syl 20	. . . . . . . . . . 11 ⊢ (x = z → (x ‘∪dom x) = (x ‘∪dom z))
13		fveq1 3729	. . . . . . . . . . 11 ⊢ (x = z → (x ‘∪dom z) = (z ‘∪dom z))
14	12, 13	eqtrd 1510	. . . . . . . . . 10 ⊢ (x = z → (x ‘∪dom x) = (z ‘∪dom z))
15	14	fveq2d 3734	. . . . . . . . 9 ⊢ (x = z → (H ‘(x ‘∪dom x)) = (H ‘(z ‘∪dom z)))
16	15	eqeq2d 1489	. . . . . . . 8 ⊢ (x = z → (y = (H ‘(x ‘∪dom x)) ↔ y = (H ‘(z ‘∪dom z))))
17	9, 16	anbi12d 630	. . . . . . 7 ⊢ (x = z → ((¬ (x = ∅ ⋁ Lim dom x) ⋀ y = (H ‘(x ‘∪dom x))) ↔ (¬ (z = ∅ ⋁ Lim dom z) ⋀ y = (H ‘(z ‘∪dom z)))))
18		rneq 3345	. . . . . . . . . 10 ⊢ (x = z → ran x = ran z)
19	18	unieqd 2516	. . . . . . . . 9 ⊢ (x = z → ∪ran x = ∪ran z)
20	19	eqeq2d 1489	. . . . . . . 8 ⊢ (x = z → (y = ∪ran x ↔ y = ∪ran z))
21	7, 20	anbi12d 630	. . . . . . 7 ⊢ (x = z → ((Lim dom x ⋀ y = ∪ran x) ↔ (Lim dom z ⋀ y = ∪ran z)))
22	4, 17, 21	3orbi123d 894	. . . . . 6 ⊢ (x = z → (((x = ∅ ⋀ y = A) ⋁ (¬ (x = ∅ ⋁ Lim dom x) ⋀ y = (H ‘(x ‘∪dom x))) ⋁ (Lim dom x ⋀ y = ∪ran x)) ↔ ((z = ∅ ⋀ y = A) ⋁ (¬ (z = ∅ ⋁ Lim dom z) ⋀ y = (H ‘(z ‘∪dom z))) ⋁ (Lim dom z ⋀ y = ∪ran z))))
23	2, 22	anbi12d 630	. . . . 5 ⊢ (x = z → ((w = ⟨x, y⟩ ⋀ ((x = ∅ ⋀ y = A) ⋁ (¬ (x = ∅ ⋁ Lim dom x) ⋀ y = (H ‘(x ‘∪dom x))) ⋁ (Lim dom x ⋀ y = ∪ran x))) ↔ (w = ⟨z, y⟩ ⋀ ((z = ∅ ⋀ y = A) ⋁ (¬ (z = ∅ ⋁ Lim dom z) ⋀ y = (H ‘(z ‘∪dom z))) ⋁ (Lim dom z ⋀ y = ∪ran z)))))
24	23	exbidv 1281	. . . 4 ⊢ (x = z → (∃y(w = ⟨x, y⟩ ⋀ ((x = ∅ ⋀ y = A) ⋁ (¬ (x = ∅ ⋁ Lim dom x) ⋀ y = (H ‘(x ‘∪dom x))) ⋁ (Lim dom x ⋀ y = ∪ran x))) ↔ ∃y(w = ⟨z, y⟩ ⋀ ((z = ∅ ⋀ y = A) ⋁ (¬ (z = ∅ ⋁ Lim dom z) ⋀ y = (H ‘(z ‘∪dom z))) ⋁ (Lim dom z ⋀ y = ∪ran z)))))
25	24	cbvexv 1317	. . 3 ⊢ (∃x∃y(w = ⟨x, y⟩ ⋀ ((x = ∅ ⋀ y = A) ⋁ (¬ (x = ∅ ⋁ Lim dom x) ⋀ y = (H ‘(x ‘∪dom x))) ⋁ (Lim dom x ⋀ y = ∪ran x))) ↔ ∃z∃y(w = ⟨z, y⟩ ⋀ ((z = ∅ ⋀ y = A) ⋁ (¬ (z = ∅ ⋁ Lim dom z) ⋀ y = (H ‘(z ‘∪dom z))) ⋁ (Lim dom z ⋀ y = ∪ran z))))
26	25	abbii 1578	. 2 ⊢ {w∣∃x∃y(w = ⟨x, y⟩ ⋀ ((x = ∅ ⋀ y = A) ⋁ (¬ (x = ∅ ⋁ Lim dom x) ⋀ y = (H ‘(x ‘∪dom x))) ⋁ (Lim dom x ⋀ y = ∪ran x)))} = {w∣∃z∃y(w = ⟨z, y⟩ ⋀ ((z = ∅ ⋀ y = A) ⋁ (¬ (z = ∅ ⋁ Lim dom z) ⋀ y = (H ‘(z ‘∪dom z))) ⋁ (Lim dom z ⋀ y = ∪ran z)))}
27		df-opab 2672	. 2 ⊢ {⟨x, y⟩∣((x = ∅ ⋀ y = A) ⋁ (¬ (x = ∅ ⋁ Lim dom x) ⋀ y = (H ‘(x ‘∪dom x))) ⋁ (Lim dom x ⋀ y = ∪ran x))} = {w∣∃x∃y(w = ⟨x, y⟩ ⋀ ((x = ∅ ⋀ y = A) ⋁ (¬ (x = ∅ ⋁ Lim dom x) ⋀ y = (H ‘(x ‘∪dom x))) ⋁ (Lim dom x ⋀ y = ∪ran x)))}
28		df-opab 2672	. 2 ⊢ {⟨z, y⟩∣((z = ∅ ⋀ y = A) ⋁ (¬ (z = ∅ ⋁ Lim dom z) ⋀ y = (H ‘(z ‘∪dom z))) ⋁ (Lim dom z ⋀ y = ∪ran z))} = {w∣∃z∃y(w = ⟨z, y⟩ ⋀ ((z = ∅ ⋀ y = A) ⋁ (¬ (z = ∅ ⋁ Lim dom z) ⋀ y = (H ‘(z ‘∪dom z))) ⋁ (Lim dom z ⋀ y = ∪ran z)))}
29	26, 27, 28	3eqtr4 1508	1 ⊢ {⟨x, y⟩∣((x = ∅ ⋀ y = A) ⋁ (¬ (x = ∅ ⋁ Lim dom x) ⋀ y = (H ‘(x ‘∪dom x))) ⋁ (Lim dom x ⋀ y = ∪ran x))} = {⟨z, y⟩∣((z = ∅ ⋀ y = A) ⋁ (¬ (z = ∅ ⋁ Lim dom z) ⋀ y = (H ‘(z ‘∪dom z))) ⋁ (Lim dom z ⋀ y = ∪ran z))}

Syntax hints:  ¬ wn 2   ↔ wb 146   ⋁ wo 222   ⋀ wa 223   ⋁ w3o 776   = wceq 958  ∃wex 982  {cab 1466  ∅c0 2283  ⟨cop 2415  ∪cuni 2507  {copab 2671  Lim wlim 2955  dom cdm 3176  ran crn 3177   ‘cfv 3188

This theorem is referenced by:  rdgval 3946  rdg0 3947  rdgsuc 3948  rdglim 3949

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-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-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  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-tr 2686  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-lim 2959  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fv 3204

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