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Theorem fun11iun 5090
Description: The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by Mario Carneiro, 20-May-2013.) (Revised by Mario Carneiro, 24-Jun-2015.)
Hypotheses
Ref Expression
fun11iun.1 (x = yB = 𝐶)
fun11iun.2 B V
Assertion
Ref Expression
fun11iun (x A (B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) → x A B: x A 𝐷1-1𝑆)
Distinct variable groups:   x,A   y,A   y,B   x,𝐶   x,𝑆
Allowed substitution hints:   B(x)   𝐶(y)   𝐷(x,y)   𝑆(y)

Proof of Theorem fun11iun
Dummy variables u v z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . . . . . . . 10 u V
2 eqeq1 2043 . . . . . . . . . . 11 (z = u → (z = Bu = B))
32rexbidv 2321 . . . . . . . . . 10 (z = u → (x A z = Bx A u = B))
41, 3elab 2681 . . . . . . . . 9 (u {zx A z = B} ↔ x A u = B)
5 r19.29 2444 . . . . . . . . . 10 ((x A (B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) x A u = B) → x A ((B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) u = B))
6 nfv 1418 . . . . . . . . . . . 12 x(Fun u Fun u)
7 nfre1 2359 . . . . . . . . . . . . . 14 xx A z = B
87nfab 2179 . . . . . . . . . . . . 13 x{zx A z = B}
9 nfv 1418 . . . . . . . . . . . . 13 x(uv vu)
108, 9nfralxy 2354 . . . . . . . . . . . 12 xv {zx A z = B} (uv vu)
116, 10nfan 1454 . . . . . . . . . . 11 x((Fun u Fun u) v {zx A z = B} (uv vu))
12 f1eq1 5030 . . . . . . . . . . . . . . . 16 (u = B → (u:𝐷1-1𝑆B:𝐷1-1𝑆))
1312biimparc 283 . . . . . . . . . . . . . . 15 ((B:𝐷1-1𝑆 u = B) → u:𝐷1-1𝑆)
14 df-f1 4850 . . . . . . . . . . . . . . . 16 (u:𝐷1-1𝑆 ↔ (u:𝐷𝑆 Fun u))
15 ffun 4991 . . . . . . . . . . . . . . . . 17 (u:𝐷𝑆 → Fun u)
1615anim1i 323 . . . . . . . . . . . . . . . 16 ((u:𝐷𝑆 Fun u) → (Fun u Fun u))
1714, 16sylbi 114 . . . . . . . . . . . . . . 15 (u:𝐷1-1𝑆 → (Fun u Fun u))
1813, 17syl 14 . . . . . . . . . . . . . 14 ((B:𝐷1-1𝑆 u = B) → (Fun u Fun u))
1918adantlr 446 . . . . . . . . . . . . 13 (((B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) u = B) → (Fun u Fun u))
20 vex 2554 . . . . . . . . . . . . . . . 16 v V
21 eqeq1 2043 . . . . . . . . . . . . . . . . 17 (z = v → (z = Bv = B))
2221rexbidv 2321 . . . . . . . . . . . . . . . 16 (z = v → (x A z = Bx A v = B))
2320, 22elab 2681 . . . . . . . . . . . . . . 15 (v {zx A z = B} ↔ x A v = B)
24 fun11iun.1 . . . . . . . . . . . . . . . . . 18 (x = yB = 𝐶)
2524eqeq2d 2048 . . . . . . . . . . . . . . . . 17 (x = y → (v = Bv = 𝐶))
2625cbvrexv 2528 . . . . . . . . . . . . . . . 16 (x A v = By A v = 𝐶)
27 r19.29 2444 . . . . . . . . . . . . . . . . . . 19 ((y A (B𝐶 𝐶B) y A v = 𝐶) → y A ((B𝐶 𝐶B) v = 𝐶))
28 sseq12 2962 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((u = B v = 𝐶) → (uvB𝐶))
2928ancoms 255 . . . . . . . . . . . . . . . . . . . . . . . 24 ((v = 𝐶 u = B) → (uvB𝐶))
30 sseq12 2962 . . . . . . . . . . . . . . . . . . . . . . . 24 ((v = 𝐶 u = B) → (vu𝐶B))
3129, 30orbi12d 706 . . . . . . . . . . . . . . . . . . . . . . 23 ((v = 𝐶 u = B) → ((uv vu) ↔ (B𝐶 𝐶B)))
3231biimprcd 149 . . . . . . . . . . . . . . . . . . . . . 22 ((B𝐶 𝐶B) → ((v = 𝐶 u = B) → (uv vu)))
3332expdimp 246 . . . . . . . . . . . . . . . . . . . . 21 (((B𝐶 𝐶B) v = 𝐶) → (u = B → (uv vu)))
3433rexlimivw 2423 . . . . . . . . . . . . . . . . . . . 20 (y A ((B𝐶 𝐶B) v = 𝐶) → (u = B → (uv vu)))
3534imp 115 . . . . . . . . . . . . . . . . . . 19 ((y A ((B𝐶 𝐶B) v = 𝐶) u = B) → (uv vu))
3627, 35sylan 267 . . . . . . . . . . . . . . . . . 18 (((y A (B𝐶 𝐶B) y A v = 𝐶) u = B) → (uv vu))
3736an32s 502 . . . . . . . . . . . . . . . . 17 (((y A (B𝐶 𝐶B) u = B) y A v = 𝐶) → (uv vu))
3837adantlll 449 . . . . . . . . . . . . . . . 16 ((((B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) u = B) y A v = 𝐶) → (uv vu))
3926, 38sylan2b 271 . . . . . . . . . . . . . . 15 ((((B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) u = B) x A v = B) → (uv vu))
4023, 39sylan2b 271 . . . . . . . . . . . . . 14 ((((B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) u = B) v {zx A z = B}) → (uv vu))
4140ralrimiva 2386 . . . . . . . . . . . . 13 (((B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) u = B) → v {zx A z = B} (uv vu))
4219, 41jca 290 . . . . . . . . . . . 12 (((B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) u = B) → ((Fun u Fun u) v {zx A z = B} (uv vu)))
4342a1i 9 . . . . . . . . . . 11 (x A → (((B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) u = B) → ((Fun u Fun u) v {zx A z = B} (uv vu))))
4411, 43rexlimi 2420 . . . . . . . . . 10 (x A ((B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) u = B) → ((Fun u Fun u) v {zx A z = B} (uv vu)))
455, 44syl 14 . . . . . . . . 9 ((x A (B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) x A u = B) → ((Fun u Fun u) v {zx A z = B} (uv vu)))
464, 45sylan2b 271 . . . . . . . 8 ((x A (B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) u {zx A z = B}) → ((Fun u Fun u) v {zx A z = B} (uv vu)))
4746ralrimiva 2386 . . . . . . 7 (x A (B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) → u {zx A z = B} ((Fun u Fun u) v {zx A z = B} (uv vu)))
48 fun11uni 4912 . . . . . . 7 (u {zx A z = B} ((Fun u Fun u) v {zx A z = B} (uv vu)) → (Fun {zx A z = B} Fun {zx A z = B}))
4947, 48syl 14 . . . . . 6 (x A (B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) → (Fun {zx A z = B} Fun {zx A z = B}))
5049simpld 105 . . . . 5 (x A (B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) → Fun {zx A z = B})
51 fun11iun.2 . . . . . . 7 B V
5251dfiun2 3682 . . . . . 6 x A B = {zx A z = B}
5352funeqi 4865 . . . . 5 (Fun x A B ↔ Fun {zx A z = B})
5450, 53sylibr 137 . . . 4 (x A (B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) → Fun x A B)
55 nfra1 2349 . . . . . . 7 xx A (B:𝐷1-1𝑆 y A (B𝐶 𝐶B))
56 rsp 2363 . . . . . . . . 9 (x A (B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) → (x A → (B:𝐷1-1𝑆 y A (B𝐶 𝐶B))))
571eldm2 4476 . . . . . . . . . . 11 (u dom Bvu, v B)
58 f1dm 5039 . . . . . . . . . . . 12 (B:𝐷1-1𝑆 → dom B = 𝐷)
5958eleq2d 2104 . . . . . . . . . . 11 (B:𝐷1-1𝑆 → (u dom Bu 𝐷))
6057, 59syl5bbr 183 . . . . . . . . . 10 (B:𝐷1-1𝑆 → (vu, v Bu 𝐷))
6160adantr 261 . . . . . . . . 9 ((B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) → (vu, v Bu 𝐷))
6256, 61syl6 29 . . . . . . . 8 (x A (B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) → (x A → (vu, v Bu 𝐷)))
6362imp 115 . . . . . . 7 ((x A (B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) x A) → (vu, v Bu 𝐷))
6455, 63rexbida 2315 . . . . . 6 (x A (B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) → (x A vu, v Bx A u 𝐷))
65 eliun 3652 . . . . . . . 8 (⟨u, v x A Bx Au, v B)
6665exbii 1493 . . . . . . 7 (vu, v x A Bvx Au, v B)
671eldm2 4476 . . . . . . 7 (u dom x A Bvu, v x A B)
68 rexcom4 2571 . . . . . . 7 (x A vu, v Bvx Au, v B)
6966, 67, 683bitr4i 201 . . . . . 6 (u dom x A Bx A vu, v B)
70 eliun 3652 . . . . . 6 (u x A 𝐷x A u 𝐷)
7164, 69, 703bitr4g 212 . . . . 5 (x A (B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) → (u dom x A Bu x A 𝐷))
7271eqrdv 2035 . . . 4 (x A (B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) → dom x A B = x A 𝐷)
73 df-fn 4848 . . . 4 ( x A B Fn x A 𝐷 ↔ (Fun x A B dom x A B = x A 𝐷))
7454, 72, 73sylanbrc 394 . . 3 (x A (B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) → x A B Fn x A 𝐷)
75 rniun 4677 . . . 4 ran x A B = x A ran B
76 f1f 5035 . . . . . . . 8 (B:𝐷1-1𝑆B:𝐷𝑆)
77 frn 4995 . . . . . . . 8 (B:𝐷𝑆 → ran B𝑆)
7876, 77syl 14 . . . . . . 7 (B:𝐷1-1𝑆 → ran B𝑆)
7978adantr 261 . . . . . 6 ((B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) → ran B𝑆)
8079ralimi 2378 . . . . 5 (x A (B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) → x A ran B𝑆)
81 iunss 3689 . . . . 5 ( x A ran B𝑆x A ran B𝑆)
8280, 81sylibr 137 . . . 4 (x A (B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) → x A ran B𝑆)
8375, 82syl5eqss 2983 . . 3 (x A (B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) → ran x A B𝑆)
84 df-f 4849 . . 3 ( x A B: x A 𝐷𝑆 ↔ ( x A B Fn x A 𝐷 ran x A B𝑆))
8574, 83, 84sylanbrc 394 . 2 (x A (B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) → x A B: x A 𝐷𝑆)
8649simprd 107 . . 3 (x A (B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) → Fun {zx A z = B})
8752cnveqi 4453 . . . 4 x A B = {zx A z = B}
8887funeqi 4865 . . 3 (Fun x A B ↔ Fun {zx A z = B})
8986, 88sylibr 137 . 2 (x A (B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) → Fun x A B)
90 df-f1 4850 . 2 ( x A B: x A 𝐷1-1𝑆 ↔ ( x A B: x A 𝐷𝑆 Fun x A B))
9185, 89, 90sylanbrc 394 1 (x A (B:𝐷1-1𝑆 y A (B𝐶 𝐶B)) → x A B: x A 𝐷1-1𝑆)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wo 628   = wceq 1242  wex 1378   wcel 1390  {cab 2023  wral 2300  wrex 2301  Vcvv 2551  wss 2911  cop 3370   cuni 3571   ciun 3648  ccnv 4287  dom cdm 4288  ran crn 4289  Fun wfun 4839   Fn wfn 4840  wf 4841  1-1wf1 4842
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850
This theorem is referenced by: (None)
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