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Theorem fmpt2x 5927
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
fmpt2x.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
fmpt2x (∀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 fmpt2x
Dummy variables 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2618 . . . . . . . 8 𝑧 ∈ V
2 vex 2618 . . . . . . . 8 𝑤 ∈ V
31, 2op1std 5876 . . . . . . 7 (𝑣 = ⟨𝑧, 𝑤⟩ → (1st𝑣) = 𝑧)
43csbeq1d 2928 . . . . . 6 (𝑣 = ⟨𝑧, 𝑤⟩ → (1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶 = 𝑧 / 𝑥(2nd𝑣) / 𝑦𝐶)
51, 2op2ndd 5877 . . . . . . . 8 (𝑣 = ⟨𝑧, 𝑤⟩ → (2nd𝑣) = 𝑤)
65csbeq1d 2928 . . . . . . 7 (𝑣 = ⟨𝑧, 𝑤⟩ → (2nd𝑣) / 𝑦𝐶 = 𝑤 / 𝑦𝐶)
76csbeq2dv 2945 . . . . . 6 (𝑣 = ⟨𝑧, 𝑤⟩ → 𝑧 / 𝑥(2nd𝑣) / 𝑦𝐶 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
84, 7eqtrd 2117 . . . . 5 (𝑣 = ⟨𝑧, 𝑤⟩ → (1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
98eleq1d 2153 . . . 4 (𝑣 = ⟨𝑧, 𝑤⟩ → ((1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶𝐷𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷))
109raliunxp 4545 . . 3 (∀𝑣 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)(1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶𝐷 ↔ ∀𝑧𝐴𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷)
11 nfv 1464 . . . . . . 7 𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)
12 nfv 1464 . . . . . . 7 𝑤((𝑥𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)
13 nfv 1464 . . . . . . . . 9 𝑥 𝑧𝐴
14 nfcsb1v 2952 . . . . . . . . . 10 𝑥𝑧 / 𝑥𝐵
1514nfcri 2219 . . . . . . . . 9 𝑥 𝑤𝑧 / 𝑥𝐵
1613, 15nfan 1500 . . . . . . . 8 𝑥(𝑧𝐴𝑤𝑧 / 𝑥𝐵)
17 nfcsb1v 2952 . . . . . . . . 9 𝑥𝑧 / 𝑥𝑤 / 𝑦𝐶
1817nfeq2 2236 . . . . . . . 8 𝑥 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶
1916, 18nfan 1500 . . . . . . 7 𝑥((𝑧𝐴𝑤𝑧 / 𝑥𝐵) ∧ 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
20 nfv 1464 . . . . . . . 8 𝑦(𝑧𝐴𝑤𝑧 / 𝑥𝐵)
21 nfcv 2225 . . . . . . . . . 10 𝑦𝑧
22 nfcsb1v 2952 . . . . . . . . . 10 𝑦𝑤 / 𝑦𝐶
2321, 22nfcsb 2954 . . . . . . . . 9 𝑦𝑧 / 𝑥𝑤 / 𝑦𝐶
2423nfeq2 2236 . . . . . . . 8 𝑦 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶
2520, 24nfan 1500 . . . . . . 7 𝑦((𝑧𝐴𝑤𝑧 / 𝑥𝐵) ∧ 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
26 eleq1 2147 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
2726adantr 270 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑥𝐴𝑧𝐴))
28 eleq1 2147 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑦𝐵𝑤𝐵))
29 csbeq1a 2930 . . . . . . . . . . 11 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
3029eleq2d 2154 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑤𝐵𝑤𝑧 / 𝑥𝐵))
3128, 30sylan9bbr 451 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑦𝐵𝑤𝑧 / 𝑥𝐵))
3227, 31anbi12d 457 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝐴𝑦𝐵) ↔ (𝑧𝐴𝑤𝑧 / 𝑥𝐵)))
33 csbeq1a 2930 . . . . . . . . . 10 (𝑦 = 𝑤𝐶 = 𝑤 / 𝑦𝐶)
34 csbeq1a 2930 . . . . . . . . . 10 (𝑥 = 𝑧𝑤 / 𝑦𝐶 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
3533, 34sylan9eqr 2139 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
3635eqeq2d 2096 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑣 = 𝐶𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶))
3732, 36anbi12d 457 . . . . . . 7 ((𝑥 = 𝑧𝑦 = 𝑤) → (((𝑥𝐴𝑦𝐵) ∧ 𝑣 = 𝐶) ↔ ((𝑧𝐴𝑤𝑧 / 𝑥𝐵) ∧ 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)))
3811, 12, 19, 25, 37cbvoprab12 5679 . . . . . 6 {⟨⟨𝑥, 𝑦⟩, 𝑣⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)} = {⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∣ ((𝑧𝐴𝑤𝑧 / 𝑥𝐵) ∧ 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)}
39 df-mpt2 5618 . . . . . 6 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑣⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)}
40 df-mpt2 5618 . . . . . 6 (𝑧𝐴, 𝑤𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶) = {⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∣ ((𝑧𝐴𝑤𝑧 / 𝑥𝐵) ∧ 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)}
4138, 39, 403eqtr4i 2115 . . . . 5 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶)
42 fmpt2x.1 . . . . 5 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
438mpt2mptx 5696 . . . . 5 (𝑣 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵) ↦ (1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶) = (𝑧𝐴, 𝑤𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶)
4441, 42, 433eqtr4i 2115 . . . 4 𝐹 = (𝑣 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵) ↦ (1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶)
4544fmpt 5412 . . 3 (∀𝑣 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)(1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶𝐷𝐹: 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)⟶𝐷)
4610, 45bitr3i 184 . 2 (∀𝑧𝐴𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷𝐹: 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)⟶𝐷)
47 nfv 1464 . . 3 𝑧𝑦𝐵 𝐶𝐷
4817nfel1 2235 . . . 4 𝑥𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷
4914, 48nfralxy 2410 . . 3 𝑥𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷
50 nfv 1464 . . . . 5 𝑤 𝐶𝐷
5122nfel1 2235 . . . . 5 𝑦𝑤 / 𝑦𝐶𝐷
5233eleq1d 2153 . . . . 5 (𝑦 = 𝑤 → (𝐶𝐷𝑤 / 𝑦𝐶𝐷))
5350, 51, 52cbvral 2582 . . . 4 (∀𝑦𝐵 𝐶𝐷 ↔ ∀𝑤𝐵 𝑤 / 𝑦𝐶𝐷)
5434eleq1d 2153 . . . . 5 (𝑥 = 𝑧 → (𝑤 / 𝑦𝐶𝐷𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷))
5529, 54raleqbidv 2570 . . . 4 (𝑥 = 𝑧 → (∀𝑤𝐵 𝑤 / 𝑦𝐶𝐷 ↔ ∀𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷))
5653, 55syl5bb 190 . . 3 (𝑥 = 𝑧 → (∀𝑦𝐵 𝐶𝐷 ↔ ∀𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷))
5747, 49, 56cbvral 2582 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 ↔ ∀𝑧𝐴𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷)
58 nfcv 2225 . . . 4 𝑧({𝑥} × 𝐵)
59 nfcv 2225 . . . . 5 𝑥{𝑧}
6059, 14nfxp 4437 . . . 4 𝑥({𝑧} × 𝑧 / 𝑥𝐵)
61 sneq 3442 . . . . 5 (𝑥 = 𝑧 → {𝑥} = {𝑧})
6261, 29xpeq12d 4436 . . . 4 (𝑥 = 𝑧 → ({𝑥} × 𝐵) = ({𝑧} × 𝑧 / 𝑥𝐵))
6358, 60, 62cbviun 3750 . . 3 𝑥𝐴 ({𝑥} × 𝐵) = 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)
6463feq2i 5120 . 2 (𝐹: 𝑥𝐴 ({𝑥} × 𝐵)⟶𝐷𝐹: 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)⟶𝐷)
6546, 57, 643bitr4i 210 1 (∀𝑥𝐴𝑦𝐵 𝐶𝐷𝐹: 𝑥𝐴 ({𝑥} × 𝐵)⟶𝐷)
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1287  wcel 1436  wral 2355  csb 2922  {csn 3431  cop 3434   ciun 3713  cmpt 3874   × cxp 4409  wf 4977  cfv 4981  {coprab 5614  cmpt2 5615  1st c1st 5866  2nd c2nd 5867
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-un 4234
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-id 4094  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-fv 4989  df-oprab 5617  df-mpt2 5618  df-1st 5868  df-2nd 5869
This theorem is referenced by:  fmpt2  5928
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