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Theorem fnmpt2ovd 7488
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
fnmpt2ovd.m (𝜑𝑀 Fn (𝐴 × 𝐵))
fnmpt2ovd.s ((𝑖 = 𝑎𝑗 = 𝑏) → 𝐷 = 𝐶)
fnmpt2ovd.d ((𝜑𝑖𝐴𝑗𝐵) → 𝐷𝑈)
fnmpt2ovd.c ((𝜑𝑎𝐴𝑏𝐵) → 𝐶𝑉)
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   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 fnmpt2ovd
StepHypRef Expression
1 fnmpt2ovd.m . . 3 (𝜑𝑀 Fn (𝐴 × 𝐵))
2 fnmpt2ovd.c . . . . . 6 ((𝜑𝑎𝐴𝑏𝐵) → 𝐶𝑉)
323expb 1150 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑏𝐵)) → 𝐶𝑉)
43ralrimivva 3152 . . . 4 (𝜑 → ∀𝑎𝐴𝑏𝐵 𝐶𝑉)
5 eqid 2799 . . . . 5 (𝑎𝐴, 𝑏𝐵𝐶) = (𝑎𝐴, 𝑏𝐵𝐶)
65fnmpt2 7474 . . . 4 (∀𝑎𝐴𝑏𝐵 𝐶𝑉 → (𝑎𝐴, 𝑏𝐵𝐶) Fn (𝐴 × 𝐵))
74, 6syl 17 . . 3 (𝜑 → (𝑎𝐴, 𝑏𝐵𝐶) Fn (𝐴 × 𝐵))
8 eqfnov2 7001 . . 3 ((𝑀 Fn (𝐴 × 𝐵) ∧ (𝑎𝐴, 𝑏𝐵𝐶) Fn (𝐴 × 𝐵)) → (𝑀 = (𝑎𝐴, 𝑏𝐵𝐶) ↔ ∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = (𝑖(𝑎𝐴, 𝑏𝐵𝐶)𝑗)))
91, 7, 8syl2anc 580 . 2 (𝜑 → (𝑀 = (𝑎𝐴, 𝑏𝐵𝐶) ↔ ∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = (𝑖(𝑎𝐴, 𝑏𝐵𝐶)𝑗)))
10 nfcv 2941 . . . . . . . 8 𝑎𝐷
11 nfcv 2941 . . . . . . . 8 𝑏𝐷
12 nfcv 2941 . . . . . . . 8 𝑖𝐶
13 nfcv 2941 . . . . . . . 8 𝑗𝐶
14 fnmpt2ovd.s . . . . . . . 8 ((𝑖 = 𝑎𝑗 = 𝑏) → 𝐷 = 𝐶)
1510, 11, 12, 13, 14cbvmpt2 6968 . . . . . . 7 (𝑖𝐴, 𝑗𝐵𝐷) = (𝑎𝐴, 𝑏𝐵𝐶)
1615eqcomi 2808 . . . . . 6 (𝑎𝐴, 𝑏𝐵𝐶) = (𝑖𝐴, 𝑗𝐵𝐷)
1716a1i 11 . . . . 5 (𝜑 → (𝑎𝐴, 𝑏𝐵𝐶) = (𝑖𝐴, 𝑗𝐵𝐷))
1817oveqd 6895 . . . 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 3170 . 2 (𝜑 → (∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = (𝑖(𝑎𝐴, 𝑏𝐵𝐶)𝑗) ↔ ∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗)))
21 simprl 788 . . . . 5 ((𝜑 ∧ (𝑖𝐴𝑗𝐵)) → 𝑖𝐴)
22 simprr 790 . . . . 5 ((𝜑 ∧ (𝑖𝐴𝑗𝐵)) → 𝑗𝐵)
23 fnmpt2ovd.d . . . . . 6 ((𝜑𝑖𝐴𝑗𝐵) → 𝐷𝑈)
24233expb 1150 . . . . 5 ((𝜑 ∧ (𝑖𝐴𝑗𝐵)) → 𝐷𝑈)
25 eqid 2799 . . . . . 6 (𝑖𝐴, 𝑗𝐵𝐷) = (𝑖𝐴, 𝑗𝐵𝐷)
2625ovmpt4g 7017 . . . . 5 ((𝑖𝐴𝑗𝐵𝐷𝑈) → (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗) = 𝐷)
2721, 22, 24, 26syl3anc 1491 . . . 4 ((𝜑 ∧ (𝑖𝐴𝑗𝐵)) → (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗) = 𝐷)
2827eqeq2d 2809 . . 3 ((𝜑 ∧ (𝑖𝐴𝑗𝐵)) → ((𝑖𝑀𝑗) = (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗) ↔ (𝑖𝑀𝑗) = 𝐷))
29282ralbidva 3169 . 2 (𝜑 → (∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗) ↔ ∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = 𝐷))
309, 20, 293bitrd 297 1 (𝜑 → (𝑀 = (𝑎𝐴, 𝑏𝐵𝐶) ↔ ∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3089   × cxp 5310   Fn wfn 6096  (class class class)co 6878  cmpt2 6880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402
This theorem is referenced by:  mpt2frlmd  20441
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