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Theorem fmpt2x 7188
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 3192 . . . . . . . 8 𝑧 ∈ V
2 vex 3192 . . . . . . . 8 𝑤 ∈ V
31, 2op1std 7130 . . . . . . 7 (𝑣 = ⟨𝑧, 𝑤⟩ → (1st𝑣) = 𝑧)
43csbeq1d 3525 . . . . . 6 (𝑣 = ⟨𝑧, 𝑤⟩ → (1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶 = 𝑧 / 𝑥(2nd𝑣) / 𝑦𝐶)
51, 2op2ndd 7131 . . . . . . . 8 (𝑣 = ⟨𝑧, 𝑤⟩ → (2nd𝑣) = 𝑤)
65csbeq1d 3525 . . . . . . 7 (𝑣 = ⟨𝑧, 𝑤⟩ → (2nd𝑣) / 𝑦𝐶 = 𝑤 / 𝑦𝐶)
76csbeq2dv 3969 . . . . . 6 (𝑣 = ⟨𝑧, 𝑤⟩ → 𝑧 / 𝑥(2nd𝑣) / 𝑦𝐶 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
84, 7eqtrd 2655 . . . . 5 (𝑣 = ⟨𝑧, 𝑤⟩ → (1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
98eleq1d 2683 . . . 4 (𝑣 = ⟨𝑧, 𝑤⟩ → ((1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶𝐷𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷))
109raliunxp 5226 . . 3 (∀𝑣 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)(1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶𝐷 ↔ ∀𝑧𝐴𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷)
11 nfv 1840 . . . . . . 7 𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)
12 nfv 1840 . . . . . . 7 𝑤((𝑥𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)
13 nfv 1840 . . . . . . . . 9 𝑥 𝑧𝐴
14 nfcsb1v 3534 . . . . . . . . . 10 𝑥𝑧 / 𝑥𝐵
1514nfcri 2755 . . . . . . . . 9 𝑥 𝑤𝑧 / 𝑥𝐵
1613, 15nfan 1825 . . . . . . . 8 𝑥(𝑧𝐴𝑤𝑧 / 𝑥𝐵)
17 nfcsb1v 3534 . . . . . . . . 9 𝑥𝑧 / 𝑥𝑤 / 𝑦𝐶
1817nfeq2 2776 . . . . . . . 8 𝑥 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶
1916, 18nfan 1825 . . . . . . 7 𝑥((𝑧𝐴𝑤𝑧 / 𝑥𝐵) ∧ 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
20 nfv 1840 . . . . . . . 8 𝑦(𝑧𝐴𝑤𝑧 / 𝑥𝐵)
21 nfcv 2761 . . . . . . . . . 10 𝑦𝑧
22 nfcsb1v 3534 . . . . . . . . . 10 𝑦𝑤 / 𝑦𝐶
2321, 22nfcsb 3536 . . . . . . . . 9 𝑦𝑧 / 𝑥𝑤 / 𝑦𝐶
2423nfeq2 2776 . . . . . . . 8 𝑦 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶
2520, 24nfan 1825 . . . . . . 7 𝑦((𝑧𝐴𝑤𝑧 / 𝑥𝐵) ∧ 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
26 eleq1 2686 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
2726adantr 481 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑥𝐴𝑧𝐴))
28 eleq1 2686 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑦𝐵𝑤𝐵))
29 csbeq1a 3527 . . . . . . . . . . 11 (𝑥 = 𝑧𝐵 = 𝑧 / 𝑥𝐵)
3029eleq2d 2684 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑤𝐵𝑤𝑧 / 𝑥𝐵))
3128, 30sylan9bbr 736 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑦𝐵𝑤𝑧 / 𝑥𝐵))
3227, 31anbi12d 746 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝐴𝑦𝐵) ↔ (𝑧𝐴𝑤𝑧 / 𝑥𝐵)))
33 csbeq1a 3527 . . . . . . . . . 10 (𝑦 = 𝑤𝐶 = 𝑤 / 𝑦𝐶)
34 csbeq1a 3527 . . . . . . . . . 10 (𝑥 = 𝑧𝑤 / 𝑦𝐶 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
3533, 34sylan9eqr 2677 . . . . . . . . 9 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)
3635eqeq2d 2631 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑣 = 𝐶𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶))
3732, 36anbi12d 746 . . . . . . 7 ((𝑥 = 𝑧𝑦 = 𝑤) → (((𝑥𝐴𝑦𝐵) ∧ 𝑣 = 𝐶) ↔ ((𝑧𝐴𝑤𝑧 / 𝑥𝐵) ∧ 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)))
3811, 12, 19, 25, 37cbvoprab12 6689 . . . . . 6 {⟨⟨𝑥, 𝑦⟩, 𝑣⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)} = {⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∣ ((𝑧𝐴𝑤𝑧 / 𝑥𝐵) ∧ 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)}
39 df-mpt2 6615 . . . . . 6 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑣⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑣 = 𝐶)}
40 df-mpt2 6615 . . . . . 6 (𝑧𝐴, 𝑤𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶) = {⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∣ ((𝑧𝐴𝑤𝑧 / 𝑥𝐵) ∧ 𝑣 = 𝑧 / 𝑥𝑤 / 𝑦𝐶)}
4138, 39, 403eqtr4i 2653 . . . . 5 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶)
42 fmpt2x.1 . . . . 5 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
438mpt2mptx 6711 . . . . 5 (𝑣 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵) ↦ (1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶) = (𝑧𝐴, 𝑤𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶)
4441, 42, 433eqtr4i 2653 . . . 4 𝐹 = (𝑣 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵) ↦ (1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶)
4544fmpt 6342 . . 3 (∀𝑣 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)(1st𝑣) / 𝑥(2nd𝑣) / 𝑦𝐶𝐷𝐹: 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)⟶𝐷)
4610, 45bitr3i 266 . 2 (∀𝑧𝐴𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷𝐹: 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)⟶𝐷)
47 nfv 1840 . . 3 𝑧𝑦𝐵 𝐶𝐷
4817nfel1 2775 . . . 4 𝑥𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷
4914, 48nfral 2940 . . 3 𝑥𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷
50 nfv 1840 . . . . 5 𝑤 𝐶𝐷
5122nfel1 2775 . . . . 5 𝑦𝑤 / 𝑦𝐶𝐷
5233eleq1d 2683 . . . . 5 (𝑦 = 𝑤 → (𝐶𝐷𝑤 / 𝑦𝐶𝐷))
5350, 51, 52cbvral 3158 . . . 4 (∀𝑦𝐵 𝐶𝐷 ↔ ∀𝑤𝐵 𝑤 / 𝑦𝐶𝐷)
5434eleq1d 2683 . . . . 5 (𝑥 = 𝑧 → (𝑤 / 𝑦𝐶𝐷𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷))
5529, 54raleqbidv 3144 . . . 4 (𝑥 = 𝑧 → (∀𝑤𝐵 𝑤 / 𝑦𝐶𝐷 ↔ ∀𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷))
5653, 55syl5bb 272 . . 3 (𝑥 = 𝑧 → (∀𝑦𝐵 𝐶𝐷 ↔ ∀𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷))
5747, 49, 56cbvral 3158 . 2 (∀𝑥𝐴𝑦𝐵 𝐶𝐷 ↔ ∀𝑧𝐴𝑤 𝑧 / 𝑥𝐵𝑧 / 𝑥𝑤 / 𝑦𝐶𝐷)
58 nfcv 2761 . . . 4 𝑧({𝑥} × 𝐵)
59 nfcv 2761 . . . . 5 𝑥{𝑧}
6059, 14nfxp 5107 . . . 4 𝑥({𝑧} × 𝑧 / 𝑥𝐵)
61 sneq 4163 . . . . 5 (𝑥 = 𝑧 → {𝑥} = {𝑧})
6261, 29xpeq12d 5105 . . . 4 (𝑥 = 𝑧 → ({𝑥} × 𝐵) = ({𝑧} × 𝑧 / 𝑥𝐵))
6358, 60, 62cbviun 4528 . . 3 𝑥𝐴 ({𝑥} × 𝐵) = 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)
6463feq2i 5999 . 2 (𝐹: 𝑥𝐴 ({𝑥} × 𝐵)⟶𝐷𝐹: 𝑧𝐴 ({𝑧} × 𝑧 / 𝑥𝐵)⟶𝐷)
6546, 57, 643bitr4i 292 1 (∀𝑥𝐴𝑦𝐵 𝐶𝐷𝐹: 𝑥𝐴 ({𝑥} × 𝐵)⟶𝐷)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  csb 3518  {csn 4153  cop 4159   ciun 4490  cmpt 4678   × cxp 5077  wf 5848  cfv 5852  {coprab 6611  cmpt2 6612  1st c1st 7118  2nd c2nd 7119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121
This theorem is referenced by:  fmpt2  7189  eldmcoa  16647  gsum2d2lem  18304  gsum2d2  18305  gsumcom2  18306  dmdprd  18329  dprdval  18334  dprd2d2  18375  ablfaclem2  18417  ptbasfi  21307  ptcmplem1  21779  prdsxmslem2  22257  tglnfn  25359
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