Theorem fnmpoovd 6389

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	⊢ (𝜑 → 𝑀 Fn (𝐴 × 𝐵))
fnmpoovd.s	⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → 𝐷 = 𝐶)
fnmpoovd.d	⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵) → 𝐷 ∈ 𝑈)
fnmpoovd.c	⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝐶 ∈ 𝑉)

Assertion

Ref	Expression
fnmpoovd	⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (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

Step	Hyp	Ref	Expression
1		fnmpoovd.m	. . 3 ⊢ (𝜑 → 𝑀 Fn (𝐴 × 𝐵))
2		fnmpoovd.c	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝐶 ∈ 𝑉)
3	2	3expb 1231	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵)) → 𝐶 ∈ 𝑉)
4	3	ralrimivva 2615	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝐶 ∈ 𝑉)
5		eqid 2231	. . . . 5 ⊢ (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)
6	5	fnmpo 6376	. . . 4 ⊢ (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 𝐶 ∈ 𝑉 → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) Fn (𝐴 × 𝐵))
7	4, 6	syl 14	. . 3 ⊢ (𝜑 → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) Fn (𝐴 × 𝐵))
8		eqfnov2 6139	. . 3 ⊢ ((𝑀 Fn (𝐴 × 𝐵) ∧ (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) Fn (𝐴 × 𝐵)) → (𝑀 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = (𝑖(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑗)))
9	1, 7, 8	syl2anc 411	. 2 ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = (𝑖(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑗)))
10		nfcv 2375	. . . . . . . 8 ⊢ Ⅎ𝑎𝐷
11		nfcv 2375	. . . . . . . 8 ⊢ Ⅎ𝑏𝐷
12		nfcv 2375	. . . . . . . 8 ⊢ Ⅎ𝑖𝐶
13		nfcv 2375	. . . . . . . 8 ⊢ Ⅎ𝑗𝐶
14		fnmpoovd.s	. . . . . . . 8 ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → 𝐷 = 𝐶)
15	10, 11, 12, 13, 14	cbvmpo 6110	. . . . . . 7 ⊢ (𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷) = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)
16	15	eqcomi 2235	. . . . . 6 ⊢ (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) = (𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)
17	16	a1i 9	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) = (𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷))
18	17	oveqd 6045	. . . 4 ⊢ (𝜑 → (𝑖(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑗) = (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗))
19	18	eqeq2d 2243	. . 3 ⊢ (𝜑 → ((𝑖𝑀𝑗) = (𝑖(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑗) ↔ (𝑖𝑀𝑗) = (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗)))
20	19	2ralbidv 2557	. 2 ⊢ (𝜑 → (∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = (𝑖(𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)𝑗) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗)))
21		simprl 531	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) → 𝑖 ∈ 𝐴)
22		simprr 533	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) → 𝑗 ∈ 𝐵)
23		fnmpoovd.d	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵) → 𝐷 ∈ 𝑈)
24	23	3expb 1231	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) → 𝐷 ∈ 𝑈)
25		eqid 2231	. . . . . 6 ⊢ (𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷) = (𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)
26	25	ovmpt4g 6154	. . . . 5 ⊢ ((𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵 ∧ 𝐷 ∈ 𝑈) → (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗) = 𝐷)
27	21, 22, 24, 26	syl3anc 1274	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) → (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗) = 𝐷)
28	27	eqeq2d 2243	. . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝐴 ∧ 𝑗 ∈ 𝐵)) → ((𝑖𝑀𝑗) = (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗) ↔ (𝑖𝑀𝑗) = 𝐷))
29	28	2ralbidva 2555	. 2 ⊢ (𝜑 → (∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = (𝑖(𝑖 ∈ 𝐴, 𝑗 ∈ 𝐵 ↦ 𝐷)𝑗) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = 𝐷))
30	9, 20, 29	3bitrd 214	1 ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐵 (𝑖𝑀𝑗) = 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2202  ∀wral 2511   × cxp 4729   Fn wfn 5328  (class class class)co 6028   ∈ cmpo 6030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313

This theorem is referenced by: (None)