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Theorem fmpox 6064
Description: Functionality, domain and codomain of a class given by the maps-to notation, where 𝐵(𝑥) is not constant but depends on 𝑥. (Contributed by NM, 29-Dec-2014.)
Hypothesis
Ref Expression
fmpox.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
fmpox (∀𝑥𝐴𝑦𝐵 𝐶𝐷𝐹: 𝑥𝐴 ({𝑥} × 𝐵)⟶𝐷)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐷,𝑦
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem fmpox
Dummy variables 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2661 . . . . . . . 8 𝑧 ∈ V
2 vex 2661 . . . . . . . 8 𝑤 ∈ V
31, 2op1std 6012 . . . . . . 7 (𝑣 = ⟨𝑧, 𝑤⟩ → (1st𝑣) = 𝑧)
43csbeq1d 2979 . . . . . 6 (𝑣 = ⟨𝑧, 𝑤⟩ → (1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶 = 𝑧 / 𝑥(2nd𝑣) / 𝑦𝐶)
51, 2op2ndd 6013 . . . . . . . 8 (𝑣 = ⟨𝑧, 𝑤⟩ → (2nd𝑣) = 𝑤)
65csbeq1d 2979 . . . . . . 7 (𝑣 = ⟨𝑧, 𝑤⟩ → (2nd𝑣) / 𝑦𝐶 = 𝑤 / 𝑦𝐶)
76csbeq2dv 2996 . . . . . 6 (𝑣 = ⟨𝑧, 𝑤⟩ → 𝑧 / 𝑥(2nd𝑣) / 𝑦𝐶 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
84, 7eqtrd 2148 . . . . 5 (𝑣 = ⟨𝑧, 𝑤⟩ → (1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
98eleq1d 2184 . . . 4 (𝑣 = ⟨𝑧, 𝑤⟩ → ((1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶𝐷𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷))
109raliunxp 4648 . . 3 (∀𝑣 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)(1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶𝐷 ↔ ∀𝑧𝐴𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷)
11 nfv 1491 . . . . . . 7 𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)
12 nfv 1491 . . . . . . 7 𝑤((𝑥𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)
13 nfv 1491 . . . . . . . . 9 𝑥 𝑧𝐴
14 nfcsb1v 3003 . . . . . . . . . 10 𝑥𝑧 / 𝑥𝐵
1514nfcri 2250 . . . . . . . . 9 𝑥 𝑤𝑧 / 𝑥𝐵
1613, 15nfan 1527 . . . . . . . 8 𝑥(𝑧𝐴𝑤𝑧 / 𝑥𝐵)
17 nfcsb1v 3003 . . . . . . . . 9 𝑥𝑧 / 𝑥𝑤 / 𝑦𝐶
1817nfeq2 2268 . . . . . . . 8 𝑥 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶
1916, 18nfan 1527 . . . . . . 7 𝑥((𝑧𝐴𝑤𝑧 / 𝑥𝐵) ∧ 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
20 nfv 1491 . . . . . . . 8 𝑦(𝑧𝐴𝑤𝑧 / 𝑥𝐵)
21 nfcv 2256 . . . . . . . . . 10 𝑦𝑧
22 nfcsb1v 3003 . . . . . . . . . 10 𝑦𝑤 / 𝑦𝐶
2321, 22nfcsb 3005 . . . . . . . . 9 𝑦𝑧 / 𝑥𝑤 / 𝑦𝐶
2423nfeq2 2268 . . . . . . . 8 𝑦 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶
2520, 24nfan 1527 . . . . . . 7 𝑦((𝑧𝐴𝑤𝑧 / 𝑥𝐵) ∧ 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
26 eleq1 2178 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
2726adantr 272 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑥𝐴𝑧𝐴))
28 eleq1 2178 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑦𝐵𝑤𝐵))
29 csbeq1a 2981 . . . . . . . . . . 11 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
3029eleq2d 2185 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑤𝐵𝑤𝑧 / 𝑥𝐵))
3128, 30sylan9bbr 456 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑦𝐵𝑤𝑧 / 𝑥𝐵))
3227, 31anbi12d 462 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝐴𝑦𝐵) ↔ (𝑧𝐴𝑤𝑧 / 𝑥𝐵)))
33 csbeq1a 2981 . . . . . . . . . 10 (𝑦 = 𝑤𝐶 = 𝑤 / 𝑦𝐶)
34 csbeq1a 2981 . . . . . . . . . 10 (𝑥 = 𝑧𝑤 / 𝑦𝐶 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
3533, 34sylan9eqr 2170 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
3635eqeq2d 2127 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑣 = 𝐶𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶))
3732, 36anbi12d 462 . . . . . . 7 ((𝑥 = 𝑧𝑦 = 𝑤) → (((𝑥𝐴𝑦𝐵) ∧ 𝑣 = 𝐶) ↔ ((𝑧𝐴𝑤𝑧 / 𝑥𝐵) ∧ 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)))
3811, 12, 19, 25, 37cbvoprab12 5811 . . . . . 6 {⟨⟨𝑥, 𝑦⟩, 𝑣⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)} = {⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∣ ((𝑧𝐴𝑤𝑧 / 𝑥𝐵) ∧ 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)}
39 df-mpo 5745 . . . . . 6 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑣⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)}
40 df-mpo 5745 . . . . . 6 (𝑧𝐴, 𝑤𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶) = {⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∣ ((𝑧𝐴𝑤𝑧 / 𝑥𝐵) ∧ 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)}
4138, 39, 403eqtr4i 2146 . . . . 5 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶)
42 fmpox.1 . . . . 5 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
438mpomptx 5828 . . . . 5 (𝑣 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵) ↦ (1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶) = (𝑧𝐴, 𝑤𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶)
4441, 42, 433eqtr4i 2146 . . . 4 𝐹 = (𝑣 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵) ↦ (1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶)
4544fmpt 5536 . . 3 (∀𝑣 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)(1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶𝐷𝐹: 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)⟶𝐷)
4610, 45bitr3i 185 . 2 (∀𝑧𝐴𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷𝐹: 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)⟶𝐷)
47 nfv 1491 . . 3 𝑧𝑦𝐵 𝐶𝐷
4817nfel1 2267 . . . 4 𝑥𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷
4914, 48nfralxy 2446 . . 3 𝑥𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷
50 nfv 1491 . . . . 5 𝑤 𝐶𝐷
5122nfel1 2267 . . . . 5 𝑦𝑤 / 𝑦𝐶𝐷
5233eleq1d 2184 . . . . 5 (𝑦 = 𝑤 → (𝐶𝐷𝑤 / 𝑦𝐶𝐷))
5350, 51, 52cbvral 2625 . . . 4 (∀𝑦𝐵 𝐶𝐷 ↔ ∀𝑤𝐵 𝑤 / 𝑦𝐶𝐷)
5434eleq1d 2184 . . . . 5 (𝑥 = 𝑧 → (𝑤 / 𝑦𝐶𝐷𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷))
5529, 54raleqbidv 2613 . . . 4 (𝑥 = 𝑧 → (∀𝑤𝐵 𝑤 / 𝑦𝐶𝐷 ↔ ∀𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷))
5653, 55syl5bb 191 . . 3 (𝑥 = 𝑧 → (∀𝑦𝐵 𝐶𝐷 ↔ ∀𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷))
5747, 49, 56cbvral 2625 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 ↔ ∀𝑧𝐴𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷)
58 nfcv 2256 . . . 4 𝑧({𝑥} × 𝐵)
59 nfcv 2256 . . . . 5 𝑥{𝑧}
6059, 14nfxp 4534 . . . 4 𝑥({𝑧} × 𝑧 / 𝑥𝐵)
61 sneq 3506 . . . . 5 (𝑥 = 𝑧 → {𝑥} = {𝑧})
6261, 29xpeq12d 4532 . . . 4 (𝑥 = 𝑧 → ({𝑥} × 𝐵) = ({𝑧} × 𝑧 / 𝑥𝐵))
6358, 60, 62cbviun 3818 . . 3 𝑥𝐴 ({𝑥} × 𝐵) = 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)
6463feq2i 5234 . 2 (𝐹: 𝑥𝐴 ({𝑥} × 𝐵)⟶𝐷𝐹: 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)⟶𝐷)
6546, 57, 643bitr4i 211 1 (∀𝑥𝐴𝑦𝐵 𝐶𝐷𝐹: 𝑥𝐴 ({𝑥} × 𝐵)⟶𝐷)
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1314  wcel 1463  wral 2391  csb 2973  {csn 3495  cop 3498   ciun 3781  cmpt 3957   × cxp 4505  wf 5087  cfv 5091  {coprab 5741  cmpo 5742  1st c1st 6002  2nd c2nd 6003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005
This theorem is referenced by:  fmpo  6065
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