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Theorem pw1equn 4331
 Description: A condition for a unit power class to equal a union. (Contributed by SF, 26-Jan-2015.)
Hypotheses
Ref Expression
pw1equn.1 A V
pw1equn.2 B V
Assertion
Ref Expression
pw1equn (1C = (AB) ↔ xy(C = (xy) A = 1x B = 1y))
Distinct variable groups:   x,A,y   x,B,y   x,C,y

Proof of Theorem pw1equn
StepHypRef Expression
1 unipw1 4325 . . . 4 1C = C
2 unieq 3900 . . . 4 (1C = (AB) → 1C = (AB))
31, 2syl5eqr 2399 . . 3 (1C = (AB) → C = (AB))
4 ssun1 3426 . . . . . 6 A (AB)
5 sseq2 3293 . . . . . 6 (1C = (AB) → (A 1CA (AB)))
64, 5mpbiri 224 . . . . 5 (1C = (AB) → A 1C)
7 pw1ss1c 4158 . . . . 5 1C 1c
86, 7syl6ss 3284 . . . 4 (1C = (AB) → A 1c)
9 eqpw1uni 4330 . . . 4 (A 1cA = 1A)
108, 9syl 15 . . 3 (1C = (AB) → A = 1A)
11 ssun2 3427 . . . . . 6 B (AB)
12 sseq2 3293 . . . . . 6 (1C = (AB) → (B 1CB (AB)))
1311, 12mpbiri 224 . . . . 5 (1C = (AB) → B 1C)
1413, 7syl6ss 3284 . . . 4 (1C = (AB) → B 1c)
15 eqpw1uni 4330 . . . 4 (B 1cB = 1B)
1614, 15syl 15 . . 3 (1C = (AB) → B = 1B)
17 pw1equn.1 . . . . 5 A V
1817uniex 4317 . . . 4 A V
19 pw1equn.2 . . . . 5 B V
2019uniex 4317 . . . 4 B V
21 uneq12 3413 . . . . . . 7 ((x = A y = B) → (xy) = (AB))
22 uniun 3910 . . . . . . 7 (AB) = (AB)
2321, 22syl6eqr 2403 . . . . . 6 ((x = A y = B) → (xy) = (AB))
2423eqeq2d 2364 . . . . 5 ((x = A y = B) → (C = (xy) ↔ C = (AB)))
25 pw1eq 4143 . . . . . . 7 (x = A1x = 1A)
2625eqeq2d 2364 . . . . . 6 (x = A → (A = 1xA = 1A))
2726adantr 451 . . . . 5 ((x = A y = B) → (A = 1xA = 1A))
28 pw1eq 4143 . . . . . . 7 (y = B1y = 1B)
2928eqeq2d 2364 . . . . . 6 (y = B → (B = 1yB = 1B))
3029adantl 452 . . . . 5 ((x = A y = B) → (B = 1yB = 1B))
3124, 27, 303anbi123d 1252 . . . 4 ((x = A y = B) → ((C = (xy) A = 1x B = 1y) ↔ (C = (AB) A = 1A B = 1B)))
3218, 20, 31spc2ev 2947 . . 3 ((C = (AB) A = 1A B = 1B) → xy(C = (xy) A = 1x B = 1y))
333, 10, 16, 32syl3anc 1182 . 2 (1C = (AB) → xy(C = (xy) A = 1x B = 1y))
34 pw1un 4163 . . . 4 1(xy) = (1x1y)
35 pw1eq 4143 . . . . . 6 (C = (xy) → 1C = 1(xy))
36 uneq12 3413 . . . . . 6 ((A = 1x B = 1y) → (AB) = (1x1y))
3735, 36eqeqan12d 2368 . . . . 5 ((C = (xy) (A = 1x B = 1y)) → (1C = (AB) ↔ 1(xy) = (1x1y)))
38373impb 1147 . . . 4 ((C = (xy) A = 1x B = 1y) → (1C = (AB) ↔ 1(xy) = (1x1y)))
3934, 38mpbiri 224 . . 3 ((C = (xy) A = 1x B = 1y) → 1C = (AB))
4039exlimivv 1635 . 2 (xy(C = (xy) A = 1x B = 1y) → 1C = (AB))
4133, 40impbii 180 1 (1C = (AB) ↔ xy(C = (xy) A = 1x B = 1y))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∪ cun 3207   ⊆ wss 3257  ∪cuni 3891  1cc1c 4134  ℘1cpw1 4135 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-imak 4189  df-p6 4191  df-sik 4192  df-ssetk 4193 This theorem is referenced by:  taddc  6229  letc  6231
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