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Theorem fmpox 6195
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 2740 . . . . . . . 8 𝑧 ∈ V
2 vex 2740 . . . . . . . 8 𝑤 ∈ V
31, 2op1std 6143 . . . . . . 7 (𝑣 = ⟨𝑧, 𝑤⟩ → (1st𝑣) = 𝑧)
43csbeq1d 3064 . . . . . 6 (𝑣 = ⟨𝑧, 𝑤⟩ → (1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶 = 𝑧 / 𝑥(2nd𝑣) / 𝑦𝐶)
51, 2op2ndd 6144 . . . . . . . 8 (𝑣 = ⟨𝑧, 𝑤⟩ → (2nd𝑣) = 𝑤)
65csbeq1d 3064 . . . . . . 7 (𝑣 = ⟨𝑧, 𝑤⟩ → (2nd𝑣) / 𝑦𝐶 = 𝑤 / 𝑦𝐶)
76csbeq2dv 3083 . . . . . 6 (𝑣 = ⟨𝑧, 𝑤⟩ → 𝑧 / 𝑥(2nd𝑣) / 𝑦𝐶 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
84, 7eqtrd 2210 . . . . 5 (𝑣 = ⟨𝑧, 𝑤⟩ → (1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
98eleq1d 2246 . . . 4 (𝑣 = ⟨𝑧, 𝑤⟩ → ((1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶𝐷𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷))
109raliunxp 4764 . . 3 (∀𝑣 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)(1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶𝐷 ↔ ∀𝑧𝐴𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷)
11 nfv 1528 . . . . . . 7 𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)
12 nfv 1528 . . . . . . 7 𝑤((𝑥𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)
13 nfv 1528 . . . . . . . . 9 𝑥 𝑧𝐴
14 nfcsb1v 3090 . . . . . . . . . 10 𝑥𝑧 / 𝑥𝐵
1514nfcri 2313 . . . . . . . . 9 𝑥 𝑤𝑧 / 𝑥𝐵
1613, 15nfan 1565 . . . . . . . 8 𝑥(𝑧𝐴𝑤𝑧 / 𝑥𝐵)
17 nfcsb1v 3090 . . . . . . . . 9 𝑥𝑧 / 𝑥𝑤 / 𝑦𝐶
1817nfeq2 2331 . . . . . . . 8 𝑥 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶
1916, 18nfan 1565 . . . . . . 7 𝑥((𝑧𝐴𝑤𝑧 / 𝑥𝐵) ∧ 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
20 nfv 1528 . . . . . . . 8 𝑦(𝑧𝐴𝑤𝑧 / 𝑥𝐵)
21 nfcv 2319 . . . . . . . . . 10 𝑦𝑧
22 nfcsb1v 3090 . . . . . . . . . 10 𝑦𝑤 / 𝑦𝐶
2321, 22nfcsb 3094 . . . . . . . . 9 𝑦𝑧 / 𝑥𝑤 / 𝑦𝐶
2423nfeq2 2331 . . . . . . . 8 𝑦 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶
2520, 24nfan 1565 . . . . . . 7 𝑦((𝑧𝐴𝑤𝑧 / 𝑥𝐵) ∧ 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
26 eleq1 2240 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
2726adantr 276 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑥𝐴𝑧𝐴))
28 eleq1 2240 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑦𝐵𝑤𝐵))
29 csbeq1a 3066 . . . . . . . . . . 11 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
3029eleq2d 2247 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑤𝐵𝑤𝑧 / 𝑥𝐵))
3128, 30sylan9bbr 463 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑦𝐵𝑤𝑧 / 𝑥𝐵))
3227, 31anbi12d 473 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝐴𝑦𝐵) ↔ (𝑧𝐴𝑤𝑧 / 𝑥𝐵)))
33 csbeq1a 3066 . . . . . . . . . 10 (𝑦 = 𝑤𝐶 = 𝑤 / 𝑦𝐶)
34 csbeq1a 3066 . . . . . . . . . 10 (𝑥 = 𝑧𝑤 / 𝑦𝐶 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
3533, 34sylan9eqr 2232 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
3635eqeq2d 2189 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑣 = 𝐶𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶))
3732, 36anbi12d 473 . . . . . . 7 ((𝑥 = 𝑧𝑦 = 𝑤) → (((𝑥𝐴𝑦𝐵) ∧ 𝑣 = 𝐶) ↔ ((𝑧𝐴𝑤𝑧 / 𝑥𝐵) ∧ 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)))
3811, 12, 19, 25, 37cbvoprab12 5943 . . . . . 6 {⟨⟨𝑥, 𝑦⟩, 𝑣⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)} = {⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∣ ((𝑧𝐴𝑤𝑧 / 𝑥𝐵) ∧ 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)}
39 df-mpo 5874 . . . . . 6 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑣⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)}
40 df-mpo 5874 . . . . . 6 (𝑧𝐴, 𝑤𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶) = {⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∣ ((𝑧𝐴𝑤𝑧 / 𝑥𝐵) ∧ 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)}
4138, 39, 403eqtr4i 2208 . . . . 5 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶)
42 fmpox.1 . . . . 5 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
438mpomptx 5960 . . . . 5 (𝑣 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵) ↦ (1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶) = (𝑧𝐴, 𝑤𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶)
4441, 42, 433eqtr4i 2208 . . . 4 𝐹 = (𝑣 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵) ↦ (1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶)
4544fmpt 5662 . . 3 (∀𝑣 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)(1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶𝐷𝐹: 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)⟶𝐷)
4610, 45bitr3i 186 . 2 (∀𝑧𝐴𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷𝐹: 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)⟶𝐷)
47 nfv 1528 . . 3 𝑧𝑦𝐵 𝐶𝐷
4817nfel1 2330 . . . 4 𝑥𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷
4914, 48nfralxy 2515 . . 3 𝑥𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷
50 nfv 1528 . . . . 5 𝑤 𝐶𝐷
5122nfel1 2330 . . . . 5 𝑦𝑤 / 𝑦𝐶𝐷
5233eleq1d 2246 . . . . 5 (𝑦 = 𝑤 → (𝐶𝐷𝑤 / 𝑦𝐶𝐷))
5350, 51, 52cbvral 2699 . . . 4 (∀𝑦𝐵 𝐶𝐷 ↔ ∀𝑤𝐵 𝑤 / 𝑦𝐶𝐷)
5434eleq1d 2246 . . . . 5 (𝑥 = 𝑧 → (𝑤 / 𝑦𝐶𝐷𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷))
5529, 54raleqbidv 2684 . . . 4 (𝑥 = 𝑧 → (∀𝑤𝐵 𝑤 / 𝑦𝐶𝐷 ↔ ∀𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷))
5653, 55bitrid 192 . . 3 (𝑥 = 𝑧 → (∀𝑦𝐵 𝐶𝐷 ↔ ∀𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷))
5747, 49, 56cbvral 2699 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 ↔ ∀𝑧𝐴𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷)
58 nfcv 2319 . . . 4 𝑧({𝑥} × 𝐵)
59 nfcv 2319 . . . . 5 𝑥{𝑧}
6059, 14nfxp 4650 . . . 4 𝑥({𝑧} × 𝑧 / 𝑥𝐵)
61 sneq 3602 . . . . 5 (𝑥 = 𝑧 → {𝑥} = {𝑧})
6261, 29xpeq12d 4648 . . . 4 (𝑥 = 𝑧 → ({𝑥} × 𝐵) = ({𝑧} × 𝑧 / 𝑥𝐵))
6358, 60, 62cbviun 3921 . . 3 𝑥𝐴 ({𝑥} × 𝐵) = 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)
6463feq2i 5355 . 2 (𝐹: 𝑥𝐴 ({𝑥} × 𝐵)⟶𝐷𝐹: 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)⟶𝐷)
6546, 57, 643bitr4i 212 1 (∀𝑥𝐴𝑦𝐵 𝐶𝐷𝐹: 𝑥𝐴 ({𝑥} × 𝐵)⟶𝐷)
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  csb 3057  {csn 3591  cop 3594   ciun 3884  cmpt 4061   × cxp 4621  wf 5208  cfv 5212  {coprab 5870  cmpo 5871  1st c1st 6133  2nd c2nd 6134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136
This theorem is referenced by:  fmpo  6196
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