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Theorem fnmpoovd 7768
 Description: A function with a Cartesian product as domain is a mapping with two arguments defined by its operation values. (Contributed by AV, 20-Feb-2019.) (Revised by AV, 3-Jul-2022.)
Hypotheses
Ref Expression
fnmpoovd.m (𝜑𝑀 Fn (𝐴 × 𝐵))
fnmpoovd.s ((𝑖 = 𝑎𝑗 = 𝑏) → 𝐷 = 𝐶)
fnmpoovd.d ((𝜑𝑖𝐴𝑗𝐵) → 𝐷𝑈)
fnmpoovd.c ((𝜑𝑎𝐴𝑏𝐵) → 𝐶𝑉)
Assertion
Ref Expression
fnmpoovd (𝜑 → (𝑀 = (𝑎𝐴, 𝑏𝐵𝐶) ↔ ∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = 𝐷))
Distinct variable groups:   𝐴,𝑎,𝑏,𝑖,𝑗   𝐵,𝑎,𝑏,𝑖,𝑗   𝐶,𝑖,𝑗   𝐷,𝑎,𝑏   𝑖,𝑀,𝑗   𝜑,𝑎,𝑏,𝑖,𝑗
Allowed substitution hints:   𝐶(𝑎,𝑏)   𝐷(𝑖,𝑗)   𝑈(𝑖,𝑗,𝑎,𝑏)   𝑀(𝑎,𝑏)   𝑉(𝑖,𝑗,𝑎,𝑏)

Proof of Theorem fnmpoovd
StepHypRef Expression
1 fnmpoovd.m . . 3 (𝜑𝑀 Fn (𝐴 × 𝐵))
2 fnmpoovd.c . . . . . 6 ((𝜑𝑎𝐴𝑏𝐵) → 𝐶𝑉)
323expb 1117 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → 𝐶𝑉)
43ralrimivva 3156 . . . 4 (𝜑 → ∀𝑎𝐴𝑏𝐵 𝐶𝑉)
5 eqid 2798 . . . . 5 (𝑎𝐴, 𝑏𝐵𝐶) = (𝑎𝐴, 𝑏𝐵𝐶)
65fnmpo 7752 . . . 4 (∀𝑎𝐴𝑏𝐵 𝐶𝑉 → (𝑎𝐴, 𝑏𝐵𝐶) Fn (𝐴 × 𝐵))
74, 6syl 17 . . 3 (𝜑 → (𝑎𝐴, 𝑏𝐵𝐶) Fn (𝐴 × 𝐵))
8 eqfnov2 7262 . . 3 ((𝑀 Fn (𝐴 × 𝐵) ∧ (𝑎𝐴, 𝑏𝐵𝐶) Fn (𝐴 × 𝐵)) → (𝑀 = (𝑎𝐴, 𝑏𝐵𝐶) ↔ ∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = (𝑖(𝑎𝐴, 𝑏𝐵𝐶)𝑗)))
91, 7, 8syl2anc 587 . 2 (𝜑 → (𝑀 = (𝑎𝐴, 𝑏𝐵𝐶) ↔ ∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = (𝑖(𝑎𝐴, 𝑏𝐵𝐶)𝑗)))
10 nfcv 2955 . . . . . . . 8 𝑎𝐷
11 nfcv 2955 . . . . . . . 8 𝑏𝐷
12 nfcv 2955 . . . . . . . 8 𝑖𝐶
13 nfcv 2955 . . . . . . . 8 𝑗𝐶
14 fnmpoovd.s . . . . . . . 8 ((𝑖 = 𝑎𝑗 = 𝑏) → 𝐷 = 𝐶)
1510, 11, 12, 13, 14cbvmpo 7228 . . . . . . 7 (𝑖𝐴, 𝑗𝐵𝐷) = (𝑎𝐴, 𝑏𝐵𝐶)
1615eqcomi 2807 . . . . . 6 (𝑎𝐴, 𝑏𝐵𝐶) = (𝑖𝐴, 𝑗𝐵𝐷)
1716a1i 11 . . . . 5 (𝜑 → (𝑎𝐴, 𝑏𝐵𝐶) = (𝑖𝐴, 𝑗𝐵𝐷))
1817oveqd 7153 . . . 4 (𝜑 → (𝑖(𝑎𝐴, 𝑏𝐵𝐶)𝑗) = (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗))
1918eqeq2d 2809 . . 3 (𝜑 → ((𝑖𝑀𝑗) = (𝑖(𝑎𝐴, 𝑏𝐵𝐶)𝑗) ↔ (𝑖𝑀𝑗) = (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗)))
20192ralbidv 3164 . 2 (𝜑 → (∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = (𝑖(𝑎𝐴, 𝑏𝐵𝐶)𝑗) ↔ ∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗)))
21 simprl 770 . . . . 5 ((𝜑 ∧ (𝑖𝐴𝑗𝐵)) → 𝑖𝐴)
22 simprr 772 . . . . 5 ((𝜑 ∧ (𝑖𝐴𝑗𝐵)) → 𝑗𝐵)
23 fnmpoovd.d . . . . . 6 ((𝜑𝑖𝐴𝑗𝐵) → 𝐷𝑈)
24233expb 1117 . . . . 5 ((𝜑 ∧ (𝑖𝐴𝑗𝐵)) → 𝐷𝑈)
25 eqid 2798 . . . . . 6 (𝑖𝐴, 𝑗𝐵𝐷) = (𝑖𝐴, 𝑗𝐵𝐷)
2625ovmpt4g 7278 . . . . 5 ((𝑖𝐴𝑗𝐵𝐷𝑈) → (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗) = 𝐷)
2721, 22, 24, 26syl3anc 1368 . . . 4 ((𝜑 ∧ (𝑖𝐴𝑗𝐵)) → (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗) = 𝐷)
2827eqeq2d 2809 . . 3 ((𝜑 ∧ (𝑖𝐴𝑗𝐵)) → ((𝑖𝑀𝑗) = (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗) ↔ (𝑖𝑀𝑗) = 𝐷))
29282ralbidva 3163 . 2 (𝜑 → (∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗) ↔ ∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = 𝐷))
309, 20, 293bitrd 308 1 (𝜑 → (𝑀 = (𝑎𝐴, 𝑏𝐵𝐶) ↔ ∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106   × cxp 5518   Fn wfn 6320  (class class class)co 7136   ∈ cmpo 7138 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-1st 7674  df-2nd 7675 This theorem is referenced by:  mpofrlmd  20471  fedgmullem2  31129
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