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Theorem fnmpt2ovd 7412
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.)
Hypotheses
Ref Expression
fnmpt2ovd.m (𝜑𝑀 Fn (𝐴 × 𝐵))
fnmpt2ovd.s ((𝑖 = 𝑎𝑗 = 𝑏) → 𝐷 = 𝐶)
fnmpt2ovd.d ((𝜑𝑖𝐴𝑗𝐵) → 𝐷𝑈)
fnmpt2ovd.c ((𝜑𝑎𝐴𝑏𝐵) → 𝐶𝑉)
fnmpt2ovd.v (𝜑 → (𝐴𝑋𝐵𝑌))
Assertion
Ref Expression
fnmpt2ovd (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   𝑀,𝑎,𝑏,𝑖,𝑗   𝑈,𝑎,𝑏,𝑖,𝑗   𝑉,𝑎,𝑏,𝑖,𝑗   𝑋,𝑎,𝑏,𝑖,𝑗   𝑌,𝑎,𝑏,𝑖,𝑗   𝜑,𝑎,𝑏,𝑖,𝑗
Allowed substitution hints:   𝐶(𝑎,𝑏)   𝐷(𝑖,𝑗)

Proof of Theorem fnmpt2ovd
StepHypRef Expression
1 fnmpt2ovd.m . . 3 (𝜑𝑀 Fn (𝐴 × 𝐵))
2 fnmpt2ovd.c . . . . . 6 ((𝜑𝑎𝐴𝑏𝐵) → 𝐶𝑉)
323expb 1113 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → 𝐶𝑉)
43ralrimivva 3101 . . . 4 (𝜑 → ∀𝑎𝐴𝑏𝐵 𝐶𝑉)
5 eqid 2752 . . . . 5 (𝑎𝐴, 𝑏𝐵𝐶) = (𝑎𝐴, 𝑏𝐵𝐶)
65fnmpt2 7398 . . . 4 (∀𝑎𝐴𝑏𝐵 𝐶𝑉 → (𝑎𝐴, 𝑏𝐵𝐶) Fn (𝐴 × 𝐵))
74, 6syl 17 . . 3 (𝜑 → (𝑎𝐴, 𝑏𝐵𝐶) Fn (𝐴 × 𝐵))
8 eqfnov2 6924 . . 3 ((𝑀 Fn (𝐴 × 𝐵) ∧ (𝑎𝐴, 𝑏𝐵𝐶) Fn (𝐴 × 𝐵)) → (𝑀 = (𝑎𝐴, 𝑏𝐵𝐶) ↔ ∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = (𝑖(𝑎𝐴, 𝑏𝐵𝐶)𝑗)))
91, 7, 8syl2anc 696 . 2 (𝜑 → (𝑀 = (𝑎𝐴, 𝑏𝐵𝐶) ↔ ∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = (𝑖(𝑎𝐴, 𝑏𝐵𝐶)𝑗)))
10 nfcv 2894 . . . . . . . 8 𝑎𝐷
11 nfcv 2894 . . . . . . . 8 𝑏𝐷
12 nfcv 2894 . . . . . . . 8 𝑖𝐶
13 nfcv 2894 . . . . . . . 8 𝑗𝐶
14 fnmpt2ovd.s . . . . . . . 8 ((𝑖 = 𝑎𝑗 = 𝑏) → 𝐷 = 𝐶)
1510, 11, 12, 13, 14cbvmpt2 6891 . . . . . . 7 (𝑖𝐴, 𝑗𝐵𝐷) = (𝑎𝐴, 𝑏𝐵𝐶)
1615eqcomi 2761 . . . . . 6 (𝑎𝐴, 𝑏𝐵𝐶) = (𝑖𝐴, 𝑗𝐵𝐷)
1716a1i 11 . . . . 5 (𝜑 → (𝑎𝐴, 𝑏𝐵𝐶) = (𝑖𝐴, 𝑗𝐵𝐷))
1817oveqd 6822 . . . 4 (𝜑 → (𝑖(𝑎𝐴, 𝑏𝐵𝐶)𝑗) = (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗))
1918eqeq2d 2762 . . 3 (𝜑 → ((𝑖𝑀𝑗) = (𝑖(𝑎𝐴, 𝑏𝐵𝐶)𝑗) ↔ (𝑖𝑀𝑗) = (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗)))
20192ralbidv 3119 . 2 (𝜑 → (∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = (𝑖(𝑎𝐴, 𝑏𝐵𝐶)𝑗) ↔ ∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗)))
21 simprl 811 . . . . 5 ((𝜑 ∧ (𝑖𝐴𝑗𝐵)) → 𝑖𝐴)
22 simprr 813 . . . . 5 ((𝜑 ∧ (𝑖𝐴𝑗𝐵)) → 𝑗𝐵)
23 fnmpt2ovd.d . . . . . 6 ((𝜑𝑖𝐴𝑗𝐵) → 𝐷𝑈)
24233expb 1113 . . . . 5 ((𝜑 ∧ (𝑖𝐴𝑗𝐵)) → 𝐷𝑈)
25 eqid 2752 . . . . . 6 (𝑖𝐴, 𝑗𝐵𝐷) = (𝑖𝐴, 𝑗𝐵𝐷)
2625ovmpt4g 6940 . . . . 5 ((𝑖𝐴𝑗𝐵𝐷𝑈) → (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗) = 𝐷)
2721, 22, 24, 26syl3anc 1473 . . . 4 ((𝜑 ∧ (𝑖𝐴𝑗𝐵)) → (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗) = 𝐷)
2827eqeq2d 2762 . . 3 ((𝜑 ∧ (𝑖𝐴𝑗𝐵)) → ((𝑖𝑀𝑗) = (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗) ↔ (𝑖𝑀𝑗) = 𝐷))
29282ralbidva 3118 . 2 (𝜑 → (∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗) ↔ ∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = 𝐷))
309, 20, 293bitrd 294 1 (𝜑 → (𝑀 = (𝑎𝐴, 𝑏𝐵𝐶) ↔ ∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1624  wcel 2131  wral 3042   × cxp 5256   Fn wfn 6036  (class class class)co 6805  cmpt2 6807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-1st 7325  df-2nd 7326
This theorem is referenced by:  mpt2frlmd  20310
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