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Theorem fnmpoovd 8073
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 3194 . . . 4 (𝜑 → ∀𝑎𝐴𝑏𝐵 𝐶𝑉)
5 eqid 2726 . . . . 5 (𝑎𝐴, 𝑏𝐵𝐶) = (𝑎𝐴, 𝑏𝐵𝐶)
65fnmpo 8054 . . . 4 (∀𝑎𝐴𝑏𝐵 𝐶𝑉 → (𝑎𝐴, 𝑏𝐵𝐶) Fn (𝐴 × 𝐵))
74, 6syl 17 . . 3 (𝜑 → (𝑎𝐴, 𝑏𝐵𝐶) Fn (𝐴 × 𝐵))
8 eqfnov2 7535 . . 3 ((𝑀 Fn (𝐴 × 𝐵) ∧ (𝑎𝐴, 𝑏𝐵𝐶) Fn (𝐴 × 𝐵)) → (𝑀 = (𝑎𝐴, 𝑏𝐵𝐶) ↔ ∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = (𝑖(𝑎𝐴, 𝑏𝐵𝐶)𝑗)))
91, 7, 8syl2anc 583 . 2 (𝜑 → (𝑀 = (𝑎𝐴, 𝑏𝐵𝐶) ↔ ∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = (𝑖(𝑎𝐴, 𝑏𝐵𝐶)𝑗)))
10 nfcv 2897 . . . . . . . 8 𝑎𝐷
11 nfcv 2897 . . . . . . . 8 𝑏𝐷
12 nfcv 2897 . . . . . . . 8 𝑖𝐶
13 nfcv 2897 . . . . . . . 8 𝑗𝐶
14 fnmpoovd.s . . . . . . . 8 ((𝑖 = 𝑎𝑗 = 𝑏) → 𝐷 = 𝐶)
1510, 11, 12, 13, 14cbvmpo 7499 . . . . . . 7 (𝑖𝐴, 𝑗𝐵𝐷) = (𝑎𝐴, 𝑏𝐵𝐶)
1615eqcomi 2735 . . . . . 6 (𝑎𝐴, 𝑏𝐵𝐶) = (𝑖𝐴, 𝑗𝐵𝐷)
1716a1i 11 . . . . 5 (𝜑 → (𝑎𝐴, 𝑏𝐵𝐶) = (𝑖𝐴, 𝑗𝐵𝐷))
1817oveqd 7422 . . . 4 (𝜑 → (𝑖(𝑎𝐴, 𝑏𝐵𝐶)𝑗) = (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗))
1918eqeq2d 2737 . . 3 (𝜑 → ((𝑖𝑀𝑗) = (𝑖(𝑎𝐴, 𝑏𝐵𝐶)𝑗) ↔ (𝑖𝑀𝑗) = (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗)))
20192ralbidv 3212 . 2 (𝜑 → (∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = (𝑖(𝑎𝐴, 𝑏𝐵𝐶)𝑗) ↔ ∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗)))
21 simprl 768 . . . . 5 ((𝜑 ∧ (𝑖𝐴𝑗𝐵)) → 𝑖𝐴)
22 simprr 770 . . . . 5 ((𝜑 ∧ (𝑖𝐴𝑗𝐵)) → 𝑗𝐵)
23 fnmpoovd.d . . . . . 6 ((𝜑𝑖𝐴𝑗𝐵) → 𝐷𝑈)
24233expb 1117 . . . . 5 ((𝜑 ∧ (𝑖𝐴𝑗𝐵)) → 𝐷𝑈)
25 eqid 2726 . . . . . 6 (𝑖𝐴, 𝑗𝐵𝐷) = (𝑖𝐴, 𝑗𝐵𝐷)
2625ovmpt4g 7551 . . . . 5 ((𝑖𝐴𝑗𝐵𝐷𝑈) → (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗) = 𝐷)
2721, 22, 24, 26syl3anc 1368 . . . 4 ((𝜑 ∧ (𝑖𝐴𝑗𝐵)) → (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗) = 𝐷)
2827eqeq2d 2737 . . 3 ((𝜑 ∧ (𝑖𝐴𝑗𝐵)) → ((𝑖𝑀𝑗) = (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗) ↔ (𝑖𝑀𝑗) = 𝐷))
29282ralbidva 3210 . 2 (𝜑 → (∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = (𝑖(𝑖𝐴, 𝑗𝐵𝐷)𝑗) ↔ ∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = 𝐷))
309, 20, 293bitrd 305 1 (𝜑 → (𝑀 = (𝑎𝐴, 𝑏𝐵𝐶) ↔ ∀𝑖𝐴𝑗𝐵 (𝑖𝑀𝑗) = 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3055   × cxp 5667   Fn wfn 6532  (class class class)co 7405  cmpo 7407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975
This theorem is referenced by:  mpofrlmd  21672  fedgmullem2  33233
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