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Theorem offval22 7928
Description: The function operation expressed as a mapping, variation of offval2 7553. (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)
31, 2xpexd 7601 . . 3 (𝜑 → (𝐴 × 𝐵) ∈ V)
4 xp1st 7863 . . . . 5 (𝑧 ∈ (𝐴 × 𝐵) → (1st𝑧) ∈ 𝐴)
5 xp2nd 7864 . . . . 5 (𝑧 ∈ (𝐴 × 𝐵) → (2nd𝑧) ∈ 𝐵)
64, 5jca 512 . . . 4 (𝑧 ∈ (𝐴 × 𝐵) → ((1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵))
7 fvex 6787 . . . . . 6 (2nd𝑧) ∈ V
8 fvex 6787 . . . . . 6 (1st𝑧) ∈ V
9 nfcv 2907 . . . . . . 7 𝑦(2nd𝑧)
10 nfcv 2907 . . . . . . 7 𝑥(2nd𝑧)
11 nfcv 2907 . . . . . . 7 𝑥(1st𝑧)
12 nfv 1917 . . . . . . . 8 𝑦(𝜑𝑥𝐴 ∧ (2nd𝑧) ∈ 𝐵)
13 nfcsb1v 3857 . . . . . . . . 9 𝑦(2nd𝑧) / 𝑦𝐶
1413nfel1 2923 . . . . . . . 8 𝑦(2nd𝑧) / 𝑦𝐶 ∈ V
1512, 14nfim 1899 . . . . . . 7 𝑦((𝜑𝑥𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (2nd𝑧) / 𝑦𝐶 ∈ V)
16 nfv 1917 . . . . . . . 8 𝑥(𝜑 ∧ (1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵)
17 nfcsb1v 3857 . . . . . . . . 9 𝑥(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶
1817nfel1 2923 . . . . . . . 8 𝑥(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V
1916, 18nfim 1899 . . . . . . 7 𝑥((𝜑 ∧ (1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V)
20 eleq1 2826 . . . . . . . . 9 (𝑦 = (2nd𝑧) → (𝑦𝐵 ↔ (2nd𝑧) ∈ 𝐵))
21203anbi3d 1441 . . . . . . . 8 (𝑦 = (2nd𝑧) → ((𝜑𝑥𝐴𝑦𝐵) ↔ (𝜑𝑥𝐴 ∧ (2nd𝑧) ∈ 𝐵)))
22 csbeq1a 3846 . . . . . . . . 9 (𝑦 = (2nd𝑧) → 𝐶 = (2nd𝑧) / 𝑦𝐶)
2322eleq1d 2823 . . . . . . . 8 (𝑦 = (2nd𝑧) → (𝐶 ∈ V ↔ (2nd𝑧) / 𝑦𝐶 ∈ V))
2421, 23imbi12d 345 . . . . . . 7 (𝑦 = (2nd𝑧) → (((𝜑𝑥𝐴𝑦𝐵) → 𝐶 ∈ V) ↔ ((𝜑𝑥𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (2nd𝑧) / 𝑦𝐶 ∈ V)))
25 eleq1 2826 . . . . . . . . 9 (𝑥 = (1st𝑧) → (𝑥𝐴 ↔ (1st𝑧) ∈ 𝐴))
26253anbi2d 1440 . . . . . . . 8 (𝑥 = (1st𝑧) → ((𝜑𝑥𝐴 ∧ (2nd𝑧) ∈ 𝐵) ↔ (𝜑 ∧ (1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵)))
27 csbeq1a 3846 . . . . . . . . 9 (𝑥 = (1st𝑧) → (2nd𝑧) / 𝑦𝐶 = (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
2827eleq1d 2823 . . . . . . . 8 (𝑥 = (1st𝑧) → ((2nd𝑧) / 𝑦𝐶 ∈ V ↔ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V))
2926, 28imbi12d 345 . . . . . . 7 (𝑥 = (1st𝑧) → (((𝜑𝑥𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (2nd𝑧) / 𝑦𝐶 ∈ V) ↔ ((𝜑 ∧ (1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V)))
30 offval22.c . . . . . . . 8 ((𝜑𝑥𝐴𝑦𝐵) → 𝐶𝑋)
3130elexd 3452 . . . . . . 7 ((𝜑𝑥𝐴𝑦𝐵) → 𝐶 ∈ V)
329, 10, 11, 15, 19, 24, 29, 31vtocl2gf 3508 . . . . . 6 (((2nd𝑧) ∈ V ∧ (1st𝑧) ∈ V) → ((𝜑 ∧ (1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V))
337, 8, 32mp2an 689 . . . . 5 ((𝜑 ∧ (1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V)
34333expb 1119 . . . 4 ((𝜑 ∧ ((1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵)) → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V)
356, 34sylan2 593 . . 3 ((𝜑𝑧 ∈ (𝐴 × 𝐵)) → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶 ∈ V)
36 nfcsb1v 3857 . . . . . . . . 9 𝑦(2nd𝑧) / 𝑦𝐷
3736nfel1 2923 . . . . . . . 8 𝑦(2nd𝑧) / 𝑦𝐷 ∈ V
3812, 37nfim 1899 . . . . . . 7 𝑦((𝜑𝑥𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (2nd𝑧) / 𝑦𝐷 ∈ V)
39 nfcsb1v 3857 . . . . . . . . 9 𝑥(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷
4039nfel1 2923 . . . . . . . 8 𝑥(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷 ∈ V
4116, 40nfim 1899 . . . . . . 7 𝑥((𝜑 ∧ (1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷 ∈ V)
42 csbeq1a 3846 . . . . . . . . 9 (𝑦 = (2nd𝑧) → 𝐷 = (2nd𝑧) / 𝑦𝐷)
4342eleq1d 2823 . . . . . . . 8 (𝑦 = (2nd𝑧) → (𝐷 ∈ V ↔ (2nd𝑧) / 𝑦𝐷 ∈ V))
4421, 43imbi12d 345 . . . . . . 7 (𝑦 = (2nd𝑧) → (((𝜑𝑥𝐴𝑦𝐵) → 𝐷 ∈ V) ↔ ((𝜑𝑥𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (2nd𝑧) / 𝑦𝐷 ∈ V)))
45 csbeq1a 3846 . . . . . . . . 9 (𝑥 = (1st𝑧) → (2nd𝑧) / 𝑦𝐷 = (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷)
4645eleq1d 2823 . . . . . . . 8 (𝑥 = (1st𝑧) → ((2nd𝑧) / 𝑦𝐷 ∈ V ↔ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷 ∈ V))
4726, 46imbi12d 345 . . . . . . 7 (𝑥 = (1st𝑧) → (((𝜑𝑥𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (2nd𝑧) / 𝑦𝐷 ∈ V) ↔ ((𝜑 ∧ (1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷 ∈ V)))
48 offval22.d . . . . . . . 8 ((𝜑𝑥𝐴𝑦𝐵) → 𝐷𝑌)
4948elexd 3452 . . . . . . 7 ((𝜑𝑥𝐴𝑦𝐵) → 𝐷 ∈ V)
509, 10, 11, 38, 41, 44, 47, 49vtocl2gf 3508 . . . . . 6 (((2nd𝑧) ∈ V ∧ (1st𝑧) ∈ V) → ((𝜑 ∧ (1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷 ∈ V))
517, 8, 50mp2an 689 . . . . 5 ((𝜑 ∧ (1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵) → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷 ∈ V)
52513expb 1119 . . . 4 ((𝜑 ∧ ((1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵)) → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷 ∈ V)
536, 52sylan2 593 . . 3 ((𝜑𝑧 ∈ (𝐴 × 𝐵)) → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷 ∈ V)
54 offval22.f . . . 4 (𝜑𝐹 = (𝑥𝐴, 𝑦𝐵𝐶))
55 mpompts 7905 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧 ∈ (𝐴 × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶)
5654, 55eqtrdi 2794 . . 3 (𝜑𝐹 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶))
57 offval22.g . . . 4 (𝜑𝐺 = (𝑥𝐴, 𝑦𝐵𝐷))
58 mpompts 7905 . . . 4 (𝑥𝐴, 𝑦𝐵𝐷) = (𝑧 ∈ (𝐴 × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷)
5957, 58eqtrdi 2794 . . 3 (𝜑𝐺 = (𝑧 ∈ (𝐴 × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷))
603, 35, 53, 56, 59offval2 7553 . 2 (𝜑 → (𝐹f 𝑅𝐺) = (𝑧 ∈ (𝐴 × 𝐵) ↦ ((1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑅(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷)))
61 csbov12g 7319 . . . . . . 7 ((2nd𝑧) ∈ V → (2nd𝑧) / 𝑦(𝐶𝑅𝐷) = ((2nd𝑧) / 𝑦𝐶𝑅(2nd𝑧) / 𝑦𝐷))
6261csbeq2dv 3839 . . . . . 6 ((2nd𝑧) ∈ V → (1st𝑧) / 𝑥(2nd𝑧) / 𝑦(𝐶𝑅𝐷) = (1st𝑧) / 𝑥((2nd𝑧) / 𝑦𝐶𝑅(2nd𝑧) / 𝑦𝐷))
637, 62ax-mp 5 . . . . 5 (1st𝑧) / 𝑥(2nd𝑧) / 𝑦(𝐶𝑅𝐷) = (1st𝑧) / 𝑥((2nd𝑧) / 𝑦𝐶𝑅(2nd𝑧) / 𝑦𝐷)
64 csbov12g 7319 . . . . . 6 ((1st𝑧) ∈ V → (1st𝑧) / 𝑥((2nd𝑧) / 𝑦𝐶𝑅(2nd𝑧) / 𝑦𝐷) = ((1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑅(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷))
658, 64ax-mp 5 . . . . 5 (1st𝑧) / 𝑥((2nd𝑧) / 𝑦𝐶𝑅(2nd𝑧) / 𝑦𝐷) = ((1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑅(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷)
6663, 65eqtr2i 2767 . . . 4 ((1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑅(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷) = (1st𝑧) / 𝑥(2nd𝑧) / 𝑦(𝐶𝑅𝐷)
6766mpteq2i 5179 . . 3 (𝑧 ∈ (𝐴 × 𝐵) ↦ ((1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑅(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷)) = (𝑧 ∈ (𝐴 × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦(𝐶𝑅𝐷))
68 mpompts 7905 . . 3 (𝑥𝐴, 𝑦𝐵 ↦ (𝐶𝑅𝐷)) = (𝑧 ∈ (𝐴 × 𝐵) ↦ (1st𝑧) / 𝑥(2nd𝑧) / 𝑦(𝐶𝑅𝐷))
6967, 68eqtr4i 2769 . 2 (𝑧 ∈ (𝐴 × 𝐵) ↦ ((1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐶𝑅(1st𝑧) / 𝑥(2nd𝑧) / 𝑦𝐷)) = (𝑥𝐴, 𝑦𝐵 ↦ (𝐶𝑅𝐷))
7060, 69eqtrdi 2794 1 (𝜑 → (𝐹f 𝑅𝐺) = (𝑥𝐴, 𝑦𝐵 ↦ (𝐶𝑅𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  Vcvv 3432  csb 3832  cmpt 5157   × cxp 5587  cfv 6433  (class class class)co 7275  cmpo 7277  f cof 7531  1st c1st 7829  2nd c2nd 7830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-1st 7831  df-2nd 7832
This theorem is referenced by:  matsc  21599  mdetrsca2  21753  mdetrlin2  21756  mdetunilem5  21765  smadiadetglem2  21821  mat2pmatghm  21879  pm2mpghm  21965  fedgmullem1  31710  fedgmullem2  31711
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