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Theorem fmpt2x 7125
 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 3176 . . . . . . . 8 𝑧 ∈ V
2 vex 3176 . . . . . . . 8 𝑤 ∈ V
31, 2op1std 7069 . . . . . . 7 (𝑣 = ⟨𝑧, 𝑤⟩ → (1st𝑣) = 𝑧)
43csbeq1d 3506 . . . . . 6 (𝑣 = ⟨𝑧, 𝑤⟩ → (1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶 = 𝑧 / 𝑥(2nd𝑣) / 𝑦𝐶)
51, 2op2ndd 7070 . . . . . . . 8 (𝑣 = ⟨𝑧, 𝑤⟩ → (2nd𝑣) = 𝑤)
65csbeq1d 3506 . . . . . . 7 (𝑣 = ⟨𝑧, 𝑤⟩ → (2nd𝑣) / 𝑦𝐶 = 𝑤 / 𝑦𝐶)
76csbeq2dv 3944 . . . . . 6 (𝑣 = ⟨𝑧, 𝑤⟩ → 𝑧 / 𝑥(2nd𝑣) / 𝑦𝐶 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
84, 7eqtrd 2644 . . . . 5 (𝑣 = ⟨𝑧, 𝑤⟩ → (1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
98eleq1d 2672 . . . 4 (𝑣 = ⟨𝑧, 𝑤⟩ → ((1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶𝐷𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷))
109raliunxp 5183 . . 3 (∀𝑣 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)(1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶𝐷 ↔ ∀𝑧𝐴𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷)
11 nfv 1830 . . . . . . 7 𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)
12 nfv 1830 . . . . . . 7 𝑤((𝑥𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)
13 nfv 1830 . . . . . . . . 9 𝑥 𝑧𝐴
14 nfcsb1v 3515 . . . . . . . . . 10 𝑥𝑧 / 𝑥𝐵
1514nfcri 2745 . . . . . . . . 9 𝑥 𝑤𝑧 / 𝑥𝐵
1613, 15nfan 1816 . . . . . . . 8 𝑥(𝑧𝐴𝑤𝑧 / 𝑥𝐵)
17 nfcsb1v 3515 . . . . . . . . 9 𝑥𝑧 / 𝑥𝑤 / 𝑦𝐶
1817nfeq2 2766 . . . . . . . 8 𝑥 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶
1916, 18nfan 1816 . . . . . . 7 𝑥((𝑧𝐴𝑤𝑧 / 𝑥𝐵) ∧ 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
20 nfv 1830 . . . . . . . 8 𝑦(𝑧𝐴𝑤𝑧 / 𝑥𝐵)
21 nfcv 2751 . . . . . . . . . 10 𝑦𝑧
22 nfcsb1v 3515 . . . . . . . . . 10 𝑦𝑤 / 𝑦𝐶
2321, 22nfcsb 3517 . . . . . . . . 9 𝑦𝑧 / 𝑥𝑤 / 𝑦𝐶
2423nfeq2 2766 . . . . . . . 8 𝑦 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶
2520, 24nfan 1816 . . . . . . 7 𝑦((𝑧𝐴𝑤𝑧 / 𝑥𝐵) ∧ 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
26 eleq1 2676 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
2726adantr 480 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑥𝐴𝑧𝐴))
28 eleq1 2676 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑦𝐵𝑤𝐵))
29 csbeq1a 3508 . . . . . . . . . . 11 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
3029eleq2d 2673 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑤𝐵𝑤𝑧 / 𝑥𝐵))
3128, 30sylan9bbr 733 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑦𝐵𝑤𝑧 / 𝑥𝐵))
3227, 31anbi12d 743 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝐴𝑦𝐵) ↔ (𝑧𝐴𝑤𝑧 / 𝑥𝐵)))
33 csbeq1a 3508 . . . . . . . . . 10 (𝑦 = 𝑤𝐶 = 𝑤 / 𝑦𝐶)
34 csbeq1a 3508 . . . . . . . . . 10 (𝑥 = 𝑧𝑤 / 𝑦𝐶 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
3533, 34sylan9eqr 2666 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
3635eqeq2d 2620 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑣 = 𝐶𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶))
3732, 36anbi12d 743 . . . . . . 7 ((𝑥 = 𝑧𝑦 = 𝑤) → (((𝑥𝐴𝑦𝐵) ∧ 𝑣 = 𝐶) ↔ ((𝑧𝐴𝑤𝑧 / 𝑥𝐵) ∧ 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)))
3811, 12, 19, 25, 37cbvoprab12 6627 . . . . . 6 {⟨⟨𝑥, 𝑦⟩, 𝑣⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)} = {⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∣ ((𝑧𝐴𝑤𝑧 / 𝑥𝐵) ∧ 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)}
39 df-mpt2 6554 . . . . . 6 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑣⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)}
40 df-mpt2 6554 . . . . . 6 (𝑧𝐴, 𝑤𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶) = {⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∣ ((𝑧𝐴𝑤𝑧 / 𝑥𝐵) ∧ 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)}
4138, 39, 403eqtr4i 2642 . . . . 5 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶)
42 fmpt2x.1 . . . . 5 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
438mpt2mptx 6649 . . . . 5 (𝑣 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵) ↦ (1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶) = (𝑧𝐴, 𝑤𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶)
4441, 42, 433eqtr4i 2642 . . . 4 𝐹 = (𝑣 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵) ↦ (1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶)
4544fmpt 6289 . . 3 (∀𝑣 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)(1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶𝐷𝐹: 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)⟶𝐷)
4610, 45bitr3i 265 . 2 (∀𝑧𝐴𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷𝐹: 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)⟶𝐷)
47 nfv 1830 . . 3 𝑧𝑦𝐵 𝐶𝐷
4817nfel1 2765 . . . 4 𝑥𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷
4914, 48nfral 2929 . . 3 𝑥𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷
50 nfv 1830 . . . . 5 𝑤 𝐶𝐷
5122nfel1 2765 . . . . 5 𝑦𝑤 / 𝑦𝐶𝐷
5233eleq1d 2672 . . . . 5 (𝑦 = 𝑤 → (𝐶𝐷𝑤 / 𝑦𝐶𝐷))
5350, 51, 52cbvral 3143 . . . 4 (∀𝑦𝐵 𝐶𝐷 ↔ ∀𝑤𝐵 𝑤 / 𝑦𝐶𝐷)
5434eleq1d 2672 . . . . 5 (𝑥 = 𝑧 → (𝑤 / 𝑦𝐶𝐷𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷))
5529, 54raleqbidv 3129 . . . 4 (𝑥 = 𝑧 → (∀𝑤𝐵 𝑤 / 𝑦𝐶𝐷 ↔ ∀𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷))
5653, 55syl5bb 271 . . 3 (𝑥 = 𝑧 → (∀𝑦𝐵 𝐶𝐷 ↔ ∀𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷))
5747, 49, 56cbvral 3143 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 ↔ ∀𝑧𝐴𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷)
58 nfcv 2751 . . . 4 𝑧({𝑥} × 𝐵)
59 nfcv 2751 . . . . 5 𝑥{𝑧}
6059, 14nfxp 5066 . . . 4 𝑥({𝑧} × 𝑧 / 𝑥𝐵)
61 sneq 4135 . . . . 5 (𝑥 = 𝑧 → {𝑥} = {𝑧})
6261, 29xpeq12d 5064 . . . 4 (𝑥 = 𝑧 → ({𝑥} × 𝐵) = ({𝑧} × 𝑧 / 𝑥𝐵))
6358, 60, 62cbviun 4493 . . 3 𝑥𝐴 ({𝑥} × 𝐵) = 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)
6463feq2i 5950 . 2 (𝐹: 𝑥𝐴 ({𝑥} × 𝐵)⟶𝐷𝐹: 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)⟶𝐷)
6546, 57, 643bitr4i 291 1 (∀𝑥𝐴𝑦𝐵 𝐶𝐷𝐹: 𝑥𝐴 ({𝑥} × 𝐵)⟶𝐷)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ⦋csb 3499  {csn 4125  ⟨cop 4131  ∪ ciun 4455   ↦ cmpt 4643   × cxp 5036  ⟶wf 5800  ‘cfv 5804  {coprab 6550   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060 This theorem is referenced by:  fmpt2  7126  eldmcoa  16538  gsum2d2lem  18195  gsum2d2  18196  gsumcom2  18197  dmdprd  18220  dprdval  18225  dprd2d2  18266  ablfaclem2  18308  ptbasfi  21194  ptcmplem1  21666  prdsxmslem2  22144  tglnfn  25242
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