MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  offval22 Structured version   Visualization version   GIF version

Theorem offval22 7298
Description: The function operation expressed as a mapping, variation of offval2 6956. (Contributed by SO, 15-Jul-2018.)
Hypotheses
Ref Expression
offval22.a (𝜑𝐴𝑉)
offval22.b (𝜑𝐵𝑊)
offval22.c ((𝜑𝑥𝐴𝑦𝐵) → 𝐶𝑋)
offval22.d ((𝜑𝑥𝐴𝑦𝐵) → 𝐷𝑌)
offval22.f (𝜑𝐹 = (𝑥𝐴, 𝑦𝐵𝐶))
offval22.g (𝜑𝐺 = (𝑥𝐴, 𝑦𝐵𝐷))
Assertion
Ref Expression
offval22 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐴, 𝑦𝐵 ↦ (𝐶𝑅𝐷)))
Distinct variable groups:   𝜑,𝑥,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)   𝑋(𝑥,𝑦)   𝑌(𝑥,𝑦)

Proof of Theorem offval22
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 offval22.a . . . 4 (𝜑𝐴𝑉)
2 offval22.b . . . 4 (𝜑𝐵𝑊)
3 xpexg 7002 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
41, 2, 3syl2anc 694 . . 3 (𝜑 → (𝐴 × 𝐵) ∈ V)
5 xp1st 7242 . . . . 5 (𝑧 ∈ (𝐴 × 𝐵) → (1st𝑧) ∈ 𝐴)
6 xp2nd 7243 . . . . 5 (𝑧 ∈ (𝐴 × 𝐵) → (2nd𝑧) ∈ 𝐵)
75, 6jca 553 . . . 4 (𝑧 ∈ (𝐴 × 𝐵) → ((1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵))
8 fvex 6239 . . . . . 6 (2nd𝑧) ∈ V
9 fvex 6239 . . . . . 6 (1st𝑧) ∈ V
10 nfcv 2793 . . . . . . 7 𝑦(2nd𝑧)
11 nfcv 2793 . . . . . . 7 𝑥(2nd𝑧)
12 nfcv 2793 . . . . . . 7 𝑥(1st𝑧)
13 nfv 1883 . . . . . . . 8 𝑦(𝜑𝑥𝐴 ∧ (2nd𝑧) ∈ 𝐵)
14 nfcsb1v 3582 . . . . . . . . 9 𝑦(2nd𝑧) / 𝑦𝐶
1514nfel1 2808 . . . . . . . 8 𝑦(2nd𝑧) / 𝑦𝐶 ∈ V
1613, 15nfim 1865 . . . . . . 7 𝑦((𝜑𝑥𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (2nd𝑧) / 𝑦𝐶 ∈ V)
17 nfv 1883 . . . . . . . 8 𝑥(𝜑 ∧ (1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵)
18 nfcsb1v 3582 . . . . . . . . 9 𝑥(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶
1918nfel1 2808 . . . . . . . 8 𝑥(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V
2017, 19nfim 1865 . . . . . . 7 𝑥((𝜑 ∧ (1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V)
21 eleq1 2718 . . . . . . . . 9 (𝑦 = (2nd𝑧) → (𝑦𝐵 ↔ (2nd𝑧) ∈ 𝐵))
22213anbi3d 1445 . . . . . . . 8 (𝑦 = (2nd𝑧) → ((𝜑𝑥𝐴𝑦𝐵) ↔ (𝜑𝑥𝐴 ∧ (2nd𝑧) ∈ 𝐵)))
23 csbeq1a 3575 . . . . . . . . 9 (𝑦 = (2nd𝑧) → 𝐶 = (2nd𝑧) / 𝑦𝐶)
2423eleq1d 2715 . . . . . . . 8 (𝑦 = (2nd𝑧) → (𝐶 ∈ V ↔ (2nd𝑧) / 𝑦𝐶 ∈ V))
2522, 24imbi12d 333 . . . . . . 7 (𝑦 = (2nd𝑧) → (((𝜑𝑥𝐴𝑦𝐵) → 𝐶 ∈ V) ↔ ((𝜑𝑥𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (2nd𝑧) / 𝑦𝐶 ∈ V)))
26 eleq1 2718 . . . . . . . . 9 (𝑥 = (1st𝑧) → (𝑥𝐴 ↔ (1st𝑧) ∈ 𝐴))
27263anbi2d 1444 . . . . . . . 8 (𝑥 = (1st𝑧) → ((𝜑𝑥𝐴 ∧ (2nd𝑧) ∈ 𝐵) ↔ (𝜑 ∧ (1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵)))
28 csbeq1a 3575 . . . . . . . . 9 (𝑥 = (1st𝑧) → (2nd𝑧) / 𝑦𝐶 = (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
2928eleq1d 2715 . . . . . . . 8 (𝑥 = (1st𝑧) → ((2nd𝑧) / 𝑦𝐶 ∈ V ↔ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V))
3027, 29imbi12d 333 . . . . . . 7 (𝑥 = (1st𝑧) → (((𝜑𝑥𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (2nd𝑧) / 𝑦𝐶 ∈ V) ↔ ((𝜑 ∧ (1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V)))
31 offval22.c . . . . . . . 8 ((𝜑𝑥𝐴𝑦𝐵) → 𝐶𝑋)
32 elex 3243 . . . . . . . 8 (𝐶𝑋𝐶 ∈ V)
3331, 32syl 17 . . . . . . 7 ((𝜑𝑥𝐴𝑦𝐵) → 𝐶 ∈ V)
3410, 11, 12, 16, 20, 25, 30, 33vtocl2gf 3299 . . . . . 6 (((2nd𝑧) ∈ V ∧ (1st𝑧) ∈ V) → ((𝜑 ∧ (1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V))
358, 9, 34mp2an 708 . . . . 5 ((𝜑 ∧ (1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V)
36353expb 1285 . . . 4 ((𝜑 ∧ ((1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵)) → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V)
377, 36sylan2 490 . . 3 ((𝜑𝑧 ∈ (𝐴 × 𝐵)) → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V)
38 nfcsb1v 3582 . . . . . . . . 9 𝑦(2nd𝑧) / 𝑦𝐷
3938nfel1 2808 . . . . . . . 8 𝑦(2nd𝑧) / 𝑦𝐷 ∈ V
4013, 39nfim 1865 . . . . . . 7 𝑦((𝜑𝑥𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (2nd𝑧) / 𝑦𝐷 ∈ V)
41 nfcsb1v 3582 . . . . . . . . 9 𝑥(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷
4241nfel1 2808 . . . . . . . 8 𝑥(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷 ∈ V
4317, 42nfim 1865 . . . . . . 7 𝑥((𝜑 ∧ (1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷 ∈ V)
44 csbeq1a 3575 . . . . . . . . 9 (𝑦 = (2nd𝑧) → 𝐷 = (2nd𝑧) / 𝑦𝐷)
4544eleq1d 2715 . . . . . . . 8 (𝑦 = (2nd𝑧) → (𝐷 ∈ V ↔ (2nd𝑧) / 𝑦𝐷 ∈ V))
4622, 45imbi12d 333 . . . . . . 7 (𝑦 = (2nd𝑧) → (((𝜑𝑥𝐴𝑦𝐵) → 𝐷 ∈ V) ↔ ((𝜑𝑥𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (2nd𝑧) / 𝑦𝐷 ∈ V)))
47 csbeq1a 3575 . . . . . . . . 9 (𝑥 = (1st𝑧) → (2nd𝑧) / 𝑦𝐷 = (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷)
4847eleq1d 2715 . . . . . . . 8 (𝑥 = (1st𝑧) → ((2nd𝑧) / 𝑦𝐷 ∈ V ↔ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷 ∈ V))
4927, 48imbi12d 333 . . . . . . 7 (𝑥 = (1st𝑧) → (((𝜑𝑥𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (2nd𝑧) / 𝑦𝐷 ∈ V) ↔ ((𝜑 ∧ (1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷 ∈ V)))
50 offval22.d . . . . . . . 8 ((𝜑𝑥𝐴𝑦𝐵) → 𝐷𝑌)
51 elex 3243 . . . . . . . 8 (𝐷𝑌𝐷 ∈ V)
5250, 51syl 17 . . . . . . 7 ((𝜑𝑥𝐴𝑦𝐵) → 𝐷 ∈ V)
5310, 11, 12, 40, 43, 46, 49, 52vtocl2gf 3299 . . . . . 6 (((2nd𝑧) ∈ V ∧ (1st𝑧) ∈ V) → ((𝜑 ∧ (1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷 ∈ V))
548, 9, 53mp2an 708 . . . . 5 ((𝜑 ∧ (1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷 ∈ V)
55543expb 1285 . . . 4 ((𝜑 ∧ ((1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵)) → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷 ∈ V)
567, 55sylan2 490 . . 3 ((𝜑𝑧 ∈ (𝐴 × 𝐵)) → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷 ∈ V)
57 offval22.f . . . 4 (𝜑𝐹 = (𝑥𝐴, 𝑦𝐵𝐶))
58 mpt2mpts 7279 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧 ∈ (𝐴 × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
5957, 58syl6eq 2701 . . 3 (𝜑𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶))
60 offval22.g . . . 4 (𝜑𝐺 = (𝑥𝐴, 𝑦𝐵𝐷))
61 mpt2mpts 7279 . . . 4 (𝑥𝐴, 𝑦𝐵𝐷) = (𝑧 ∈ (𝐴 × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷)
6260, 61syl6eq 2701 . . 3 (𝜑𝐺 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷))
634, 37, 56, 59, 62offval2 6956 . 2 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑧 ∈ (𝐴 × 𝐵) ↦ ((1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑅(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷)))
64 csbov12g 6729 . . . . . . 7 ((2nd𝑧) ∈ V → (2nd𝑧) / 𝑦(𝐶𝑅𝐷) = ((2nd𝑧) / 𝑦𝐶𝑅(2nd𝑧) / 𝑦𝐷))
6564csbeq2dv 4025 . . . . . 6 ((2nd𝑧) ∈ V → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦(𝐶𝑅𝐷) = (1st𝑧) / 𝑥((2nd𝑧) / 𝑦𝐶𝑅(2nd𝑧) / 𝑦𝐷))
668, 65ax-mp 5 . . . . 5 (1st𝑧) / 𝑥(2nd𝑧) / 𝑦(𝐶𝑅𝐷) = (1st𝑧) / 𝑥((2nd𝑧) / 𝑦𝐶𝑅(2nd𝑧) / 𝑦𝐷)
67 csbov12g 6729 . . . . . 6 ((1st𝑧) ∈ V → (1st𝑧) / 𝑥((2nd𝑧) / 𝑦𝐶𝑅(2nd𝑧) / 𝑦𝐷) = ((1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑅(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷))
689, 67ax-mp 5 . . . . 5 (1st𝑧) / 𝑥((2nd𝑧) / 𝑦𝐶𝑅(2nd𝑧) / 𝑦𝐷) = ((1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑅(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷)
6966, 68eqtr2i 2674 . . . 4 ((1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑅(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷) = (1st𝑧) / 𝑥(2nd𝑧) / 𝑦(𝐶𝑅𝐷)
7069mpteq2i 4774 . . 3 (𝑧 ∈ (𝐴 × 𝐵) ↦ ((1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑅(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷)) = (𝑧 ∈ (𝐴 × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦(𝐶𝑅𝐷))
71 mpt2mpts 7279 . . 3 (𝑥𝐴, 𝑦𝐵 ↦ (𝐶𝑅𝐷)) = (𝑧 ∈ (𝐴 × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦(𝐶𝑅𝐷))
7270, 71eqtr4i 2676 . 2 (𝑧 ∈ (𝐴 × 𝐵) ↦ ((1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑅(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷)) = (𝑥𝐴, 𝑦𝐵 ↦ (𝐶𝑅𝐷))
7363, 72syl6eq 2701 1 (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐴, 𝑦𝐵 ↦ (𝐶𝑅𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  Vcvv 3231  csb 3566  cmpt 4762   × cxp 5141  cfv 5926  (class class class)co 6690  cmpt2 6692  𝑓 cof 6937  1st c1st 7208  2nd c2nd 7209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-1st 7210  df-2nd 7211
This theorem is referenced by:  matsc  20304  mdetrsca2  20458  mdetrlin2  20461  mdetunilem5  20470  smadiadetglem2  20526  mat2pmatghm  20583  pm2mpghm  20669
  Copyright terms: Public domain W3C validator