Theorem xpmapenlem2 4503

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
xpmapenlem2	⊢ ((y = ⟨D, R⟩ ⋀ z ∈ C) → ((∪dom { y} ‘z) = ∪dom {(x ‘z)} ⋀ (∪ran { y} ‘z) = ∪ran {(x ‘z)}))

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 xpmapenlem2

Step	Hyp	Ref	Expression
1		sneq 2421	. . . . . . . 8 ⊢ (y = ⟨D, R⟩ → {y} = {⟨D, R⟩})
2		xpmapenlem.4	. . . . . . . . . 10 ⊢ D = {⟨z, w⟩∣(z ∈ C ⋀ w = ∪dom {(x ‘z)})}
3	2	opeq1i 2494	. . . . . . . . 9 ⊢ ⟨D, R⟩ = ⟨{⟨z, w⟩∣(z ∈ C ⋀ w = ∪dom {(x ‘z)})}, R⟩
4	3	sneqi 2422	. . . . . . . 8 ⊢ {⟨D, R⟩} = {⟨{⟨z, w⟩∣(z ∈ C ⋀ w = ∪dom {(x ‘z)})}, R⟩}
5	1, 4	syl6eq 1526	. . . . . . 7 ⊢ (y = ⟨D, R⟩ → {y} = {⟨{⟨z, w⟩∣(z ∈ C ⋀ w = ∪dom {(x ‘z)})}, R⟩})
6	5	dmeqd 3319	. . . . . 6 ⊢ (y = ⟨D, R⟩ → dom { y} = dom {⟨{⟨z, w⟩∣(z ∈ C ⋀ w = ∪dom {(x ‘z)})}, R⟩})
7	6	unieqd 2516	. . . . 5 ⊢ (y = ⟨D, R⟩ → ∪dom { y} = ∪dom {⟨{⟨z, w⟩∣(z ∈ C ⋀ w = ∪dom {(x ‘z)})}, R⟩})
8		xpmapen.3	. . . . . . 7 ⊢ C ∈ V
9	8	opabex2 3616	. . . . . 6 ⊢ {⟨z, w⟩∣(z ∈ C ⋀ w = ∪dom {(x ‘z)})} ∈ V
10	9	op1sta 3454	. . . . 5 ⊢ ∪dom {⟨{⟨z, w⟩∣(z ∈ C ⋀ w = ∪dom {(x ‘z)})}, R⟩} = {⟨z, w⟩∣(z ∈ C ⋀ w = ∪dom {(x ‘z)})}
11	7, 10	syl6eq 1526	. . . 4 ⊢ (y = ⟨D, R⟩ → ∪dom { y} = {⟨z, w⟩∣(z ∈ C ⋀ w = ∪dom {(x ‘z)})})
12	11	fveq1d 3732	. . 3 ⊢ (y = ⟨D, R⟩ → (∪dom { y} ‘z) = ({⟨z, w⟩∣(z ∈ C ⋀ w = ∪dom {(x ‘z)})} ‘z))
13		snex 2756	. . . . . 6 ⊢ {(x ‘z)} ∈ V
14	13	dmex 3366	. . . . 5 ⊢ dom {(x ‘z)} ∈ V
15	14	uniex 2876	. . . 4 ⊢ ∪dom {(x ‘z)} ∈ V
16		fvopab2 3797	. . . 4 ⊢ ((z ∈ C ⋀ ∪dom {(x ‘z)} ∈ V) → ({⟨z, w⟩∣(z ∈ C ⋀ w = ∪dom {(x ‘z)})} ‘z) = ∪dom {(x ‘z)})
17	15, 16	mpan2 698	. . 3 ⊢ (z ∈ C → ({⟨z, w⟩∣(z ∈ C ⋀ w = ∪dom {(x ‘z)})} ‘z) = ∪dom {(x ‘z)})
18	12, 17	sylan9eq 1530	. 2 ⊢ ((y = ⟨D, R⟩ ⋀ z ∈ C) → (∪dom { y} ‘z) = ∪dom {(x ‘z)})
19		xpmapenlem.5	. . . . . . . . . 10 ⊢ R = {⟨z, w⟩∣(z ∈ C ⋀ w = ∪ran {(x ‘z)})}
20	19	opeq2i 2495	. . . . . . . . 9 ⊢ ⟨D, R⟩ = ⟨D, {⟨z, w⟩∣(z ∈ C ⋀ w = ∪ran {(x ‘z)})}⟩
21	20	sneqi 2422	. . . . . . . 8 ⊢ {⟨D, R⟩} = {⟨D, {⟨z, w⟩∣(z ∈ C ⋀ w = ∪ran {(x ‘z)})}⟩}
22	1, 21	syl6eq 1526	. . . . . . 7 ⊢ (y = ⟨D, R⟩ → {y} = {⟨D, {⟨z, w⟩∣(z ∈ C ⋀ w = ∪ran {(x ‘z)})}⟩})
23	22	rneqd 3347	. . . . . 6 ⊢ (y = ⟨D, R⟩ → ran { y} = ran {⟨D, {⟨z, w⟩∣(z ∈ C ⋀ w = ∪ran {(x ‘z)})}⟩})
24	23	unieqd 2516	. . . . 5 ⊢ (y = ⟨D, R⟩ → ∪ran { y} = ∪ran {⟨D, {⟨z, w⟩∣(z ∈ C ⋀ w = ∪ran {(x ‘z)})}⟩})
25	8, 2	fopabex2 3618	. . . . . 6 ⊢ D ∈ V
26	8	opabex2 3616	. . . . . 6 ⊢ {⟨z, w⟩∣(z ∈ C ⋀ w = ∪ran {(x ‘z)})} ∈ V
27	25, 26	op2nda 3458	. . . . 5 ⊢ ∪ran {⟨D, {⟨z, w⟩∣(z ∈ C ⋀ w = ∪ran {(x ‘z)})}⟩} = {⟨z, w⟩∣(z ∈ C ⋀ w = ∪ran {(x ‘z)})}
28	24, 27	syl6eq 1526	. . . 4 ⊢ (y = ⟨D, R⟩ → ∪ran { y} = {⟨z, w⟩∣(z ∈ C ⋀ w = ∪ran {(x ‘z)})})
29	28	fveq1d 3732	. . 3 ⊢ (y = ⟨D, R⟩ → (∪ran { y} ‘z) = ({⟨z, w⟩∣(z ∈ C ⋀ w = ∪ran {(x ‘z)})} ‘z))
30	13	rnex 3367	. . . . 5 ⊢ ran {(x ‘z)} ∈ V
31	30	uniex 2876	. . . 4 ⊢ ∪ran {(x ‘z)} ∈ V
32		fvopab2 3797	. . . 4 ⊢ ((z ∈ C ⋀ ∪ran {(x ‘z)} ∈ V) → ({⟨z, w⟩∣(z ∈ C ⋀ w = ∪ran {(x ‘z)})} ‘z) = ∪ran {(x ‘z)})
33	31, 32	mpan2 698	. . 3 ⊢ (z ∈ C → ({⟨z, w⟩∣(z ∈ C ⋀ w = ∪ran {(x ‘z)})} ‘z) = ∪ran {(x ‘z)})
34	29, 33	sylan9eq 1530	. 2 ⊢ ((y = ⟨D, R⟩ ⋀ z ∈ C) → (∪ran { y} ‘z) = ∪ran {(x ‘z)})
35	18, 34	jca 288	1 ⊢ ((y = ⟨D, R⟩ ⋀ z ∈ C) → ((∪dom { y} ‘z) = ∪dom {(x ‘z)} ⋀ (∪ran { y} ‘z) = ∪ran {(x ‘z)}))

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 958   ∈ wcel 960  Vcvv 1814  {csn 2413  ⟨cop 2415  ∪cuni 2507  {copab 2671  dom cdm 3176  ran crn 3177   ‘cfv 3188

This theorem is referenced by:  xpmapenlem3 4504  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-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-fv 3204

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